I haven't owned either, but hear many good things. Honda's usually come with the higher price tag, which you can expect from the brand badge.
The GP160 equivalent (or close to) power wise in the late Briggs range is the 750, and I don't believe it is available here with the 5/8 keyway shaft option. Only 3/4 (keyway), or 5/8 (threaded / pump) - which is no good for us in these applications anyway.
I do highly recommend (and own) the B&S 550 range. They're well priced, efficient, quiet, smooth, easy to work on and in my opinion the perfect power match for the Model 45. Pretty certain there is a 3/4 keyway shaft option available with this model also.
The engine replacement can certainly assist with chassis vibration. Although it's equally (if not more) important to inspect the engine side clutch half, the clutch cone, and main/PTO shaft keyways, and even all of the individual keys. If they are looking pretty worse for wear and key fitment is loose and sloppy, do away with them and replace them. The most expensive parts out of the whole PTO assembly are going to be the main clutch cone and the engine side half. The PTO shaft is pretty cheap, and usually come supplied with keys.
Edit - BB has beat me to it!
