This is a question that does come up every so often when someone with non-HEMA experience discusses the idea of setting up a HEMA club. Of course it seems like quite a reasonable idea to continue teaching the non-HEMA material with which you are familiar, and there are probably techniques from your previous training that would be useful in various situations in HEMA sparring.
I worked through this process myself, several years ago, when I started looking at HEMA after spending around fourteen years studying karate. I had achieved my 3rd dan black belt in karate, and I thought that importing some karate techniques and concepts would help to shore up any of the many deficiencies I perceived in the HEMA systems I was trying to learn.
However, with more experience of HEMA now, I can see quite clearly that the biggest and most important deficiency was my own lack of skill at the systems I was trying to learn! Now I know that these systems can deal with almost any problem (within the appropriate context) if I apply the techniques and concepts properly – and if I need to solve a problem in a different context, I just use a different (and more appropriate) system.