I have no simple solution to this problem.
Most cellists I know keep their mute on the strings for their orchestra cello, usually close to the tailpiece, were the effect is minimum. Of course, having a large wolf-eliminator blocks this possibility, as you describe, because you have to remove it from the strings and then put it back on the bridge.
But, unless your wolf eliminator is difficult to put on and remove, you may have noticed that it modifies your sound too, and in a larger measure probably than the mute does. In fact, your mute below the bridge might be just adding a little weight to the eliminator, muffling certain notes even more.
An ironic thing is that I have seen at least one cellist using a modified loaded Tourte mute as an adjustable wolf-eliminator.
I don't know if your wolf-eliminator has been installed by a luthier, and different cellos have different needs and response to a given model, but there has been progress on cello-wolf-eliminators such as the lup-x model (which is small and not intrusive), the new harmony (very easy to put on or off) or the new magnetic (and sadly, expensive) krentz and rexz, which are not installed on the strings but on the body of the cello itself (so it would solve your problem).
Unless you play in orchestra passages with frequent use and removal of the mute, you might simply go for storing it near your rosin cake, put it on the partition holder and grab it when needed, but you don't need me to tell you that.