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Why do you think our doe is experiencing a false pregnancy? Any possibility it's for real?
A doe pulls fur as a response to a surge of hormones. This could be from pregnancy, a false pregnancy, or simply the hormones related to their normal cycle (and in spite of what some folks may tell you, rabbits do have a cycle, it's just not that obvious). Most does will hit a peak right after kindling, so in a way, fur pulling is a good indicator that you can get a doe bred.
If this doe is giving you all the right signals (which can include grunting, chinning, lifting when being petted, fur pulling, moodiness, and a dark pink to cherry red vulva lining), she should be cycling. I say "should," because a doe that is overweight may not be experiencing a wide enough range in hormone levels to cycle normally. If weight is the issue, giving the doe less feed and more exercise may help.
It is wintertime; rabbits can respond to shorter days by losing interest in breeding (wild rabbits only breed during the warm months, after all). If you don't want to wait until the longer days of Spring time, you might try essentially increasing her day length with artificial lighting.
One thing that can be done with a doe that simply won't lift when everything else is right, is forced breeding. Basically, the doe is put on a table or other flat surface, the doe is held still with one hand, and her backside lifted with the other. The buck is then allowed to do his thing. This may or may not work. The doe may struggle so violently that the only thing that is achieved is a bunch of scratches on your arms. The buck may not welcome your involvement, and be unwilling to do his part. And, even if all goes well with the actual breeding, the doe may not conceive - I know of one study that got about an 80% conception rate with natural breeding, but only about 20% when the does were force-bred.
I suspect you may be beginning to understand why I say that anyone who believes in the expression "breeds like rabbits" has never tried it."
Good luck!