Hi MacGee,
As Ceasar said, if the strap is synthetic fibre you should have no problems sealing the cut ends by passing a naked flame (i use a lighter, stops me fingers burning) over them - that type of material will melt & seal quite well just in close proximity to the flame (move the flame around as the ends start to melt). You can use your fingers to 'crimp' the edges whilst they're still warm. I've used this method for years on camera & binocular straps, it works. Just don't make the ends too big/thick to fit through the binocular strap holders!
You can always practice first on the pieces you cut off!
Cheers & Best Wishes,
Mook