you already have the right general idea, when you don’t constrain the fillet has lots of room and is perfect (but may look fat or adjacent elements may bump together) and if you constrain, the fillet has to move inward and if the element is not wide enough to contain two fillets (there will be a fillet on the other side too) it looks bad as the two fillets overlap so your only option is to cut back on the fillet radius.
You can try adding the fillet by hand with the bevel tool (which has a limit option) but you will probably get the best results by modifying the splines in Illustrator to be narrower so you can use the fillet without constraining (if you need that much radius)