There are people that run solely off convertors for years with problems but there are even more that have horror stories about burning up the boards on their trolling motors. A lot would ride on the convertor. There are single stage, 3 stage, and 4 stage convertors that I'm aware of. The multi-stage chargers have varying output voltages depending on the state of the battery charge. If there's no battery, then the charger depends on the draw from the troller to determine which phase or stage it should be in. If one is determined to run solely off convertors, the better choice would be the single stage convertor providing 13.4 to 13.6 volts.
Losing power in weeds? Check for fishing line in troller, bad battery cells, & inadequate wire size to start with. You might try putting your batteries on the bow and wire your troller direct. If it still happens that eliminates your wiring. Use a digital voltmeter to make sure your charged batteries have a voltage of about 13 volts or just above. A battery with a dead cell will have a voltage of around 9 or 10. Anything between normal and a dead cell may indicate that your battery is sulfated. Nicer chargers have a sulfation mode to burn that sulfation off and recondition your battery. If all that checks out, check out the voltages during heavy use like starting a motor. if it drops to zero or close to it, you may have an open cell that is only revealing itself when the battery heats up under draw.
If the batteries check out, it's time to get your troller looked at.