Probably my earlier speculation was correct. If you connect a wire from the main terminal to the push-on, it will turn on the internal solenoid whenever the main terminal is powered--so, you haven't exactly bypassed it, but the same thing happens. That way, you can use the original solenoid and connect the starter as you did the original one. I guess people like to do that, maybe because it's simple and you can use that convenient pushbutton on the original solenoid to turn over the engine when you're working on it. As for me, I think I would prefer to use the starter's internal solenoid.
If it were me, I would first try connecting a heavy cable from the battery to the starter; disconnect the battery and put the car in neutral, of course, while doing that. Then, reconnect the battery, and if all is well (i.e., nothing happens), connect a lighter wire from the push-on terminal and touch it to the battery (keep your fingers clear, as it will probably create a spark). The starter should then run. If it does, disconnect the battery again and make the connection to the ignition switch. Easiest way is to splice it to the wire that actuates the original solenoid (1/4" push-on terminal on its side).
The connection to the TR4's electrical system is made at the solenoid (fairly heavy brown wire), so if you remove the original solenoid, that has to be reconnected to the battery some way.
I really think this will work. Just be careful, keep the transmission in neutral, and disconnect the battery while making changes. If something gets hot or makes big sparks, stop right there.