Actually it can work quite well on one ticket. I thought you wanted a stay in MIA too which might've made it difficult but isn't too bad anyway.

Here's one reasonable option on an AA N fare. Interestingly this is allowing the surface sector NYC-WAS on a single fare component BGI-ABZ which I think is quite unusual on a cheap fare.

It involves outbound via MIA instead of LGW though. I'll see if anything can be done through LGW for a similar price.