First question: does you mare know how to accept any sort of bit pressure???
If she knows how to give when you apply pressure to the reins that she is most likely just claustrophobic to the pressure of the two fixed side reins. Horses who are more sensitive/reactive tend to be this way when you "fix" them in a certain position because it makes them feel trapped, and as we all know side reins don't release unless the horse puts it head DOWN. If you mare is pulling up on the sidereins how can we expect there to be any release...and the more she backs up and the tenser she gets, the more pressure is put on her mouth.
My mare is exactly the same, in fact I wouldn't dare put her into two fixed sidereins again until she is further on in her training.
What you need to try is only applying the OUTSIDE siderein and then string your longe line through the ring of her bit on the inside (only in a snaffle bit) and then hook the longe line snap back to the surcingle (or d-ring on you saddle) so the longe line acts as your inside siderein. This method is a much better approach because there is RELEASE. The minute you feel her tense up and want to surge backwards you release that inside longe line and let her back out...yet you will still be able to pick up on it and supple her to the inside. Like others said, longeing should have a purpose but sometimes we have to convert our methods for certain horses