As the tube base (socket) wiring diagram and most running specifications is the same for both these tubes, the real concern is the idling current (controlled by bias voltage).
The Mesa's don't have adjustable biasing and even if they did, it would be one adjustment for ALL output tubes not just per socket(s).
That said, the idle current rating of the 2 tube types needs to be the same or very close; within a few milliamps (ma.) anyway, and of course within reason for the amp.
I'm not sure what Mesa specs for the LSC, but it can't be far off from the Stiletto which is 38-40ma. You can likely get away with a lot lower or slightly higher than this range.
Just be sure to install the different types of tubes "symmetrically"; that is the outer most pair must be one type (77's OR 34's), and the inner pair, the other type. Note that this convention holds true with every amp I’ve run across and is “pretty much” standard (extremely unlikely that the Lone Star is any different).
I have modified my Mesa 50/50 power amp to run el34's and have run tubes with slightly different ratings (say, one pair at "35" and one at "39") with no problem at all. Try not to go with more of a difference with this, and preferably get the same rating.
Also, if you are at all unsure, it would be prudent to have a tech measure the idle current on each tube just to be safe.
Mixing the tube types may actually result in an interesting and unique sound quality (perhaps not unlike mixing speaker types in one cabinet….which is generally totally safe as long as the impedance is the same).
Whether your ears will like the results, or not, is subjective, but it won’t hurt the amp if the tube’s ratings are within the amp’s capabilities.
Let us know how this works out.
Good luck!