fundamentally a Presta valve is probably a better valve for a bicycle than a Schrader valve.
Given the current circumstances this may sound a little ironic, but one of the reasons I'd cite as to why Presta is better is that the pump is more likely to work more of the time with Presta than Schrader. The seal has less load on it, there is no need for a valve-opening pip or a check valve in the pump head, and small leaks are better tolerated. The valves don't fill up with crud if you leave the dustcap off, it is easier to completely deflate a tube.... I could go on.
But the fact remains that your pump presently does one valve well and the other one badly, so that may dominate your choice. I'd be a bit worried that there is some kind of lurking fault in the pump and that it may stop working altogether, though....
cheers