Not sure why a home button issue wouldn't be fixable on an SE. Unless, maybe it is a problem with the logic board? But that would usually affect way more functionality than just the home button.
If it is just the home button, though. ... You can buy a replacement home button with the flex cable for around $5, if you are willing to DIY. Just go to eBay. It's a bit of a delicate job, but it's totally doable.
Here is a video I just found:
iphone se replace home button - Google Search
For a slightly easier job, you can just buy the whole assembly for an SE -- the touch screen with the digitizer, home button and the front camera. ... for less than $20 on ebay. It's mostly the same job, but you don't have to remove the home button itself and reattach. You'd just swap out the screen.
It may be worth a try. ... for $5. The only thing is that with the new cable that doesn't match up with your logic board, the home button will work, but the touch ID won't. It's still better than having no home button.