I am of no help on a replacement alternator, however, many of the older alternators needed to get above 1500 RPMs to excite the alternator field and start charging. One of the first time I took my boat out I had previously only run it at idle in the driveway, then idles through a no wake zone and idled across the lake. Didn't want to start once we were ready to ski. Easy to check. Might also get your battery tested to eliminate it as a possible source.
Your choice 14" - 5 lug trailer wheels
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ST205/75R14 Kenda Loadstar radial trailer tire
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CARBURETOR REBUILDING - $125 + PARTS
LED trailer lights, Hubs, Spare Tire Carriers, Chrome & Powder Coated Lug Nuts