I've had a Trophy here recently with a faulty voltage regulator. The voltage across the battery would reach about 16.5 volts at 2k or 3k rpm. Clearly not regulating - just giving full output, depending only on how fast it was spinning. This will reduce battery life & if a 'wet' type it will quickly lose fluid (water) leading to plate damage (so check battery fluid level, if wet type). You need to test the battery voltage whilst revving the engine. The normal maximum is about 14.5 volts & if all is working well it will approach this with some engine revving a minute or so after starting, depending on the initial state of charge & how much effort needed to start the engine.
The regulator can be replaced without alternator removal - 3 x 8mm nuts hold the end cap in place. An aftermarket regulator (physically quite different, but functionally the same) is available around $80 ish. But sometimes an alternator can be bought for less & the regulator swapped over. Some Kawas used the same Nippondenso unit too. I found one from a ZZR1100 for $40.
N5XL's excellent info on the alternator is here:
I'll mention another fault I found on a 97 Sprint (similar electrically to a Trophy) which may be relevant. The symptoms were flickering side/pilot lights, esp at idle, & occasional cutting out when headlights were switched on (UK bike with r/handlebar headlight switch). Voltage at the battery was a little low (under 14V) but enough to suggest the alternator was working. I suspected a poor connection somewhere.
The cause was a corroded contact & some fuse body melting at the main 30 Amp fuse in the fusebox. Sorted that & fitted a new fuse & all was well.