I'll probably open myself to criticism from the experts, but I'm not a fan of putting wider wheels on the TR3, especially without the upgraded axle stubs. It changes the geometry...however minute it may be...making steering harder and putting more pressure on suspension parts than was originally designed. There, I said it, so experts blast away.
This is not intended as a "blast" in any way; I'm certainly not going to criticize anyone for wanting to run original size tires.
But I've been running wide tires (on TR6 wheels) for longer than I can remember, and I really don't see the problem. Yeah, it theoretically puts some more force on suspension parts, mostly because I buy sticky tires. (My main reason for buying wider tires is the high performance tires available.) But the suspension seems able to deal with the slight increase. Keep in mind that the front spindles and bearings are the same ones used in the 6 cylinder TRs, where they have significantly more weight on them. I've actually got a set of the "uncle jack" spindles & spacers on the shelf, haven't gotten around to installing them yet.
But steering isn't noticeably heavier; in fact I'd say it actually got easier when I swapped out the 165 tires I had on temporarily while having the car painted the first time. IMO tire compound and pressure plays a far bigger role than tread width. (I do run fairly high pressure in them, which seems to improve road handling.)
A few other points:
Earlier 3A had smaller tires, 165 will not easily fit in the spare tire compartment. (I actually carry a 145 "compact" spare because the 155s are so hard to find and I got tired of struggling to get a 165 in and out.) The compartment got bigger at TS60001, which I believe is when the 165 started.
Radial tires were available as an option fairly early on, even before the 3A was introduced. I forget the exact point offhand, but they were available on my 56.
According to Bill Piggott, there are photos of factory race cars as early as 1958 with 60 spoke wire wheels on them. As he puts it, almost certainly anyone with money could get them put on a road car. They're also listed in the later SPC as a "high speed option". So I don't think anyone could call them "wrong" for a 3A. Certainly they are much stronger, so unless you are really obsessive about being exactly original (as opposed to period correct), I'd suggest going that way. If you are replacing the wheels anyway.
I forget who offhand, but there used to be several places that could add red lines to any tire for a reasonable fee (like $30-$40 per tire). If you really want the redline look, you might see if any of them are around still. I've even heard of painting them on yourself!