Hi Bill,
If you have no cooling problems, an oil cooler might not be necessary.
If oil pressure is down (45 psi might be slightly on the low side) I'd first want to ask what oil you are using... 20W50? If so, reduced oil pressure is most likely due to some wear in the bearing surfaces, likely the mains and/or conrod big end bearings are the culprits because these have the most surface area. Simply renewing those bearing shells may be a better way to bring pressure back up. Before digging into that project, though, check the oil pressure with a second gauge to be certain you are getting accurate readings.
And, before commiting to installing an oil cooler, I'd suggest you install a gauge to monitor oil temp closely under various conditions. It should be a bit higher than the coolant temps you are seeing, probably in the range of 200-250F. Oils are designed to operate in this heat range. Overly-cooled oil can be as damaging as over-heated. Too-cool oil will give less lubrication to bearing surfaces and will not "cook off" engine impurities as well. This latter can lead to sludge in the engine and other problems. With over-heated oil the problems are also lack of lubrication due to thinning and much faster degradation of the oil.
Also keep in mind that the old coolant temp gauge might not be all that accurate. Have you double checked it lately?
However, it's true that in very hard driving situations TR2/3/4 can overheat oil and a cooler might be desirable. After checking the things listed above you might still want to install one, and there are several ways to do so.
Probably the easiest method is to just mount the oil cooler right against the front of the radiator. Done right, it won't rob much, if any, cooling effect from the radiator. It's important to keep the oil cooler right up against the front of the radiatior, not a few inches ahead of it. If there is a gap of one or two inches between the two, that can set up air turbulence that actually reduces the flow through the radiator, something to be avoided.
It's possible to mount the cooler elsewhere, hopefully some other TR3 owners will jump in with some suggestions.
On my TR4 it's getting mounted on the valance with it's own intake (similar to what's on TR250/5/6). Mounting on the valance moves it far enough forward of the radiator that I don't think it will cuase any problems with radiator air flow, plus it's an added air intake, increasing flow to both. One possible problem with valanve mounting is that the cooler is more vulnerable up front. It also needs longer hoses to reach that far.
A third common installation is inside the engine compartment, vertically oriented right next to the radiator, and with some air ducts cut through the rad shrouding to redirect some air to the cooler (for racing, some cars instead position it behind a headlight, which is left out to create an intake duct).
I think it's a bit of a tradeoff, though, whether to take some air flow away from the radiator and redirect it to an oil cooler *or* to flow the air through the cooler and then into the radiator.
Probably an adequate size oil cooler for most street TRs is a 10-row. That's what was used in the TR3 racers, once coolers were permitted. A 13-row is often what's sold by the Triumph vendors and can be used in slightly more demanding situations and/or if the engine has performance mods that generate more heat (increased compression, ofr example). A 16-row is about the largest size that might be needed on a TR. Mocal's oil coolers in these sizes happen to be almost the same width as the TR2/3/4 radiator, so are pretty easy to install using the first method mentioned above.
With any cooler, be sure to install an oil thermostat. That will help prevent over-cooling oil and the oil come up to operating temps faster. Mocal makes one now that's integral with a generic take-off sandwich plate at the oil filter. It's fully open at 185F. There are other types that go inline, but mean making 4 additional hose connections (adding to the cost of fitting and increasing the potential for leaks).
In order to install an oil cooler on a TR 4-cyl., it's common to convert to a spin-off filter first. This provides a place to put the generic sandwich plate with ports for the oil cooler hoses. Another approach is a specially made sandwich plate that goes under the TR's filter head itself. Greg Solow at the Engien Room in Santa Cruz California is the only source I know of for this type.
One concern with an oil cooler is that the routing of the oil send it to the oil filter first, then on to the cooler before being returned to the engine. This helps prevent normal engine particulates from gradlually fouling the cooler, and also helps protect it against high pressure spikes.
Besides the usual cooler, there are some other approaches to cooling the oil in a TR. One is to simply enlarge the sump pan capacity by welding on additions. The extra oil and oil sump pan surface area will do a lot to improve cooling.
Another is to install tubes through the length of the oil sump pan itself, that will allow extra air to pass through. This has the benefit of being a "passive" improvement, no moving parts to go wrong or extra plumbing connections to leak. However, there is also no way effective way to control the cooling... i.e., no thermostat... and by itself this will reduce the total volume of oil in the engine a bit. (Might be better done in conjunction with increasing the pan's capacity.)
Yet another possibility is to add an Accusump (or similar) instead. This is an accumulator for oil, which is kept under pressure and "injected" into the oiling system at startup and, say, during hard cornering if the oil pump pickup is uncovered and sucking air instead of oil. An Accusump gives an extra measure of protection. There are both manual and automatic models of these. The extra oil capacity and storage of oil in an external container will help keep oil temps under control a bit, too, although this isn't the primary purpose.
There are a few possible downsides to an oil cooler. One I've already alluded to is contamination. It's practically impossible to clean out an oil cooler and over time it will gradually get fouiled with particulates from normal engine wear. So it should be protected by filtration "upstream" and might need to be replaced every so often, perhaps every 25,000, 50,000 or 75,000 miles depending upon how the car is driven, if there is any bearing failure, etc.
Another possible concern is that it's difficult to get a complete drain and change of oil when a cooler is installed. Invariably, some oil is trapped in the lines and cooler. This can be quite alot, depending upon how the cooler is installed and how the plumbing lines are run.
Each and every plumbing connections is a potential source of leaks, and the cooler itself can be damaged by a flying rock or similar with more catastrophic results, or can even be damaged by unusually high oil pressure.
There is also a lot of cost involved with the hoses and fittings. I prefer to make up my own stainless steel braided lines and I don't really trust the normally cheaper rubber hoses (which probably were used for original installations). Doing this, you'll be using AN10 or AN12 sizes, and some of those bigger hose ends go for $25 apiece! It adds up fast!
One thing to watch out for, many of the Triumph vendors sell coolers with British pipe thread fittings, as were originally used. In the US it's harder to find hose ends to fit these, so you might choose a cooler that's got more common AN fittings instead. I've found
https://www.racerpartswholesale.com to have good prices and selection of coolers, thermostats and adapter plates.
https://www.summitracing.com or
https://www.jegs.com are good sources of the hoses and fittings.
Oh, by the way, in my opinion you might be best served by removing the pusher fan entirely and only using the puller. A pusher fan will in many cases partially block air flow. A puller fan is more efficient. Plus, removing the pusher fan will make more room for an oil cooler if you decide to mount it directly in front of the radiator.