When you mention crate engine and transmission, I'm thinking you have certain dollar figure you are willing to spend. Since you aren't THAT concerned with power output, maybe other options might be a lot cheaper (for saving $$$ for other fun stuff) and also fit better with less work and fewer headaches.
Tape measure for sure. While anything CAN be done, remember that the early cars were designed for inline engine parameters. Length of so much, width of so much, and height of so much. This same discussion came up on the Studebaker Forum with someone with a >>1950<< Champion which was the last year of the small 6 and large six. The Commander large six had longer sheet metal to accommodate the longer six. In 1951, they introduced the V8 and were able to simplify the sheet metal problem to have the same (shorter) for all models, but they had to redesign the suspension and front end to hold a wider V8. A V8 will not fit in a 1950 Studebaker car without many modifications, even though on the outside, a 50 and a 51 are virtually identical (the iconic Bulletnose)
What was suggested there I'll suggest here. For a Studebaker, changing from any early flathead 170 six to a later 185 (larger stroke in same block) six with a slight overbore and better carburation can net you 16-35 hp depending on what you do to it, and it is a direct bolt in, weighs the same, and can even be made to look entirely stock if you wish
While not exactly a guaranteed bolt in for Kaisers, remember that Checker used both Continentals and later Chev 6 ie 250 cid with the same transmission options I think, so dimensionally they should be close. In stock form, a Chev 194 was listed as 120 hp, a 230 as 140, and a 250 as 155 hp, and they are all the same block with different internal displacements. The Truck six, 292 would be even better and similar in output to the Ford 300 six.
All of these inline six engines "should" fit quite nicely in the engine bay of an early Kaiser and provide much more hp than the anemic Continental. For that matter, perhaps even one of the inline sixes from AMC/Jeep would do the trick. I have a Wagoneer 4x4 as my car towing/trailer towing vehicle, and the 4.0 L (ie 242 cid) with a 4 speed + lockup Aisin Warner automatic comes all fitted with and for A/C, power steering and brakes, and fuel injection. Besides having enough power, both engine AND that particular automatic transmission are usually completely bulletproof. A rusty Jeep could provide a complete donor drive train, and individual replacement parts are cheap at any recycler/salvage yard.