Bit of an update on this. Tried a "grippier" strap and couldn't get on with it, oddly! I like to be able to move the bass around a bit whilst I play so it just wasn't working for me. Also, whilst it helped a bit with the neck dive, I could still feel it diving...
So, got a set of Gotoh GBR640 Res-o-lites and fitted them tonight. Was a like for like swap so was dead easy, a 25 min job all in. Aaaaand it's so much better! I'm genuinely surprised what a difference it's made to be honest, and it's way nicer getting around on the bass without having to support it.
So all in, although it's cost me a fair chunk, it's improved the bass massively for me. She's a keeper anyway, so I'll get my money's worth in the long run!