Both mowers are belt drive to the transmission, although they use different belts. Not sure if the transmissions can be swapped.

However, you asked if engines can be swapped, and this appears to be be feasible. Both mowers use same deck assembly, and belt drive pulley off BBC is same for both.

Both the 3 speed and hydrostatic mowers have the same engine serial number:
GJAB and then numerals.
Does that mean the 3 speed engine could be transplanted to the hydro mower OR the hydro mower tranny transplanted to the 3 speed unit which has a dead tranny? It seems that way to me.
thanks

I just picked up a Honda HRM215 lawnmower in great condition. I put in a new paper airfilter and cleaned the carburetor and since then the engine starts like a charm but it runs for 5 - 10 seconds and then it dies. It seems to run a little longer without the paper airfilter and seems to stall earlier if the paper airfilter is installed.
Any thoughts what I have to do to get it running properly?

yes the engines between the 2 can be swapped reguardless of what tranny is in them . As for the last question about the mower that wont stay running , check the gas cap vent hole and clean it out with a paper clip or sewing needle next check fuel filter make sure its not plugged and id the correct one there are 2 different kinds one is small with just a screen for non fuel pump mowers the other is a little bigger and clear and you can see a cone shaped filter media in it that is for engines with a fuel pump and if used on a engine with out a pump will cause fuel starvation and stalling , check for fuel leaks also and if equiped check the fuel pump after those things if everything is ok then it may be time to pull the carb off and clean/rebuild it