Sounds like the alternator is able to keep up the current so that not the drive problem. The problem sounds more you simply don't have enough hp to operate the alternator and blades at the same time. The theory that it is only using about 1 hp is gong to be off as it is not a lossless system. You may be needing as much twice that or more depending how efficient the the conversion is.
As the electrical load increase so does magnetic loading from the alternator rotor that must be overcome and a GCV160 is already taxed by the grass cutting even on a regular walk behind mower sometimes. You probably will find something in the order a 240cc or larger is going to be needed; although, you could try a 190cc and see what the results are.
I know you said you don't put on a larger engine but sometime it is necessary. That what research and development is all about. It finding out what will work and what won't.
These are just my opinions so just take into consideration only as you would need to use your best judgement on what will work.
