You're going to have to match the parts for the battery, the charger, and the controller, so that they are all designed to work with the same voltages, if you want to avoid problems and workarounds.
If you use a battery that's 13s of a particular chemistry, you need ot use a charger that's setup for the full-voltage of that size and chemistry pack. In your case that's 13 x 4.2v if you're fully charging typical non-LiFePO4 lithium cells.
Otherwise, as noted, you won't get much of a charge, and you're still stuck with the same "low battery" problems, limited range, etc.
Also note that if you use a charger that is too high a voltage, then if anything goes wrong with the BMS it will overcharge the pack (and if it's badly unbalanced it could be enough overcharge to start a fire). So you want ot make sure you get one that's set for your pack's actual full-charged voltage.
Note that you could use your existing charger and BMS for charging, as long as you configure the pack for 10s to charge, but then you have to reconfigure it for 13s to use it on the bike. Doing that kind of reconfiguring on a regular basis can lead to a mistake at some point that leads to damage to battery or charger or BMS, and potentially fire. (people do this with some modular batteries, and some are successful and some aren't--accidents happen, usually when people are in a hurry or tired). It's also a PITA to have to do every time, even if nothing ever goes wrong.
The 10s BMS won't work for use on the bike during discharge, though, so you'd either have to monitor manually or get a 13s BMS for that.
Some chargers can be adjusted, but probably not as far as you need to on this one. If you want to adjust it, you'll be doing exploratory experimentation that risks damaging or destroying the charger, unless you can get the info from the manufacturer on how to do it (probably not available). Some chargers have threads on adjusting them here on ES, but I don't know that yours is one of them. Some possible threads
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