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I did use a clutch alignment tool. It slips in and out with no problem. I've tried installing the motor with the car in gear and in neutral. The through out bearing moves fine. The pressure plate is not far enough back to touch the bearing so it must be the pressure plate that is catching.

I'm trying to avoid pulling the trans because I would then have to pull the exhaust, driveshaft, shifter linkage, etc. That's my last resort if I can't get the engine on with the trans in the car.

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I hear ya there. I am just getting ready to install a 340 in a 73 Cuda and will be doing the same thing you are shortly, so this post kinda helps me too before I start my install. I have put dozens of engines and 4-speeds together and have run into all kinds of issues, some that I posted above. The pressure plate should have nothing to do with the two mating together, it is usually the clutch disc that does that. If you moved the disc after installing it with tool then the trans will not mate.