My first pair of straps were Holdfasts which were good quality but would occasionally get smashed down if I stepped on them and then I'd have to reach down and pop them out.
Switched to YNOTs which are much stiffer and never have issues. Pair these with some plastic BMX pedals and you are good to go.
The pedals you have are a better pairing with traditional straps and clips. I'd either swap the whole setup for some BMX pedals and YNOTs or get traditional clips and straps for your current pedals. Don't think you will have good results with mixing the two.