does your machine have other stitches besides the zigzag?
a button hole stitch works well to attach appliques- and yes- when you use a fusable to place your pieces they still need to be stitched in place. why does your zigzag not meet your expectations? what is wrong with it?
perhaps you could (loosen) it, make it wider, or narrower...i generally use a buttonhole stitch for machine applique..i just like it better.
some people use other decorative stitches.
as for the invisible thread- there are a number of them on the market- some melt, some don't. some stretch & break- some not so much- some can be ironed- some can not...you need to read about the one you are considering and see that it meets your needs- do know though---using invisible thread has it's own (learning curve)
put together a small practice piece-with your fusable included & try out a few different stitches, or settings until you find the one you like best- then work on the real thing...that may save you from having to remove your stitching.