I’ve finally got round to trying this with my latest roundshield, using some fairly open weave beige cloth. It works fine, with no need to paint it. Up close and straight on, you can see that there is tape under it, but from an angle, or any real distance, it just looks beigeish and not like corflute. Certainly better than the flaking, brown-painted back of my first one.
So, I’ll be doing this to all my shields from now on. It will mean changing the order I do things in, but it shouldn’t mean too many problems.
(I’m now working out whether I can retrofit to any of my existing shields. To which the answer seems to be "not unless I unpick the edges)