The advice offered in all your combined postings has really stimulated my thinking on this problem. Thanks!
I now reckon the main problem is with my trailer having built-up sides about a foot high, any straps or ropes that come over the sides are already a foot high before I secure them to any part of the mower. This means they're pulling ACROSS rather than pulling DOWN.
I think the answer might be to install some kind of floor-mounted D-rings. Attaching the tie-downs to these would pull the tractor DOWN to the floor and prevent it hopping around. What do you think?