I'd point out that if you're willing to go through test breedings and place unshowable babies in pet homes, you could breed out the vienna gene from her line.
Your best bet is to hold back an unmarked baby or two (I'm assuming you'd want does), have to reaaaaally check well for any vienna marks or signs of blue in irises like she has, and then wait until they are of breedable age. Put them with a buck that is definitely not carrying vienna and if none of the babies come out with white spots (I'd test for consistency over two or three litters before calling it for sure) then you've got two non-vienna carrying parents. The more litters that are thrown with no signs of VM, the less likely it is that your potential VC is carrying it.
The tricky thing is that marks don't always show up in carriers, so if you see any white/blue eyes in the litters at all, then unfortunately that doe is an unmarked vienna carrier, and it's back to the drawing board with trying to select a potentially non-carrier from the momma in your post.
And of course, you should always notify people of potential VC's in litters, though pet people usually like the white marks and blue eyes.
Ofc it's a project that could require a lot of holes, time, and patience, but I wanted to point out that it's doable in case you're really in love with this doe and would like to work with her line to produce anything other than BEW.
If you are looking to simply breed her (and not keep kits and breed them etc...), or if you really want to produce BEWs, I'd find a straight BEW buck if I were you (and if possible! The breeder you got her from is a good bet tho). That way each kit will have a 50/50 chance of being BEW or VM/VC. If you go this route though, there is 0% chance of a non VC kit. Any unmarked kits will definitely still carry Vienna.
Someone above mentioned showable vs unshowable, but AFAIK in Lionheads BEW is not a recognized/showable variety. So no matter what, unless you can breed out a non-VM/BEW kit they will all be unshowable. (If you get a totally unmarked kit and toenails and eyecolor are right, it CAN be shown under whatever color it appear to be, ie; Black, Blue, etc...)
I could be wrong on this but I popped over to the lionhead club page and did not see it on the list of recognized varieties. They were admitted by the ARBA fairly recently and the smoke pearl variety was officially passed just last year, so a lionhead exhibitor/Breeder might be more up to date on newly accepted varieties than me
Hope this helps! Lol everything else aside, your girl is so cute it should be illegal. I love her white nose and snip