A quick&dirty cover for the tabernacle could be any halfway sturdy, inverted plastic tub, with proper cut-outs to fit it around the mast. If that works, you can then work on a prettier solution.
Something else to consider are the jib sheets themselves. If you have two separate sheets, each attached with a bulky bowline, you are more likely to have them hang up on something than if you had a single continuous jib sheet attached with a soft shackle.
There's a thread
Rope for Various Lines where you'll find more details on ways to do that.
My setup is non-typical in that it uses a relatively heavy jib sheet. That choice was accidental and is not ideal for light wind, but it probably helps with the flapping. Making the soft shackle (or other connector) between sheet and sail from a light material may help as well.
There's finally an aspect of technique. Is she letting go of the old jib sheet too early in the tack? Keeping the jib trimmed adds power into the turn, and letting it stay backwinded even, for a short period, may help in bringing the bow around and adds time during which the crew can get the new jib ready, so that it can be sheeted in immediately.