Those are 2x6's? Not "old school" non-dimensioned 2x4's? From my experience, most lath was about 1" wide... and if that lath is 1" it would seem to scale those to 2x4.
Regardless, yes you can confidently shore up that wall by creating a temporary wall adjacent to it. If it is a basement floor it is likely to be sloped, so studs may have to be cut differently. I'd recommend attaching a top plate to the bottom of the joists, and place a "floating" bottom plate directly below on the floor. Cut the studs no more than 1/8" proud of the actual height, align the top under the floor joist, and pound in the bottom. Since it is going in at an angle, the edges will dig in some, but at only an 1/8" long it will be minimal.
You will likely have to repeat the process on the opposite side of the wall, since the joists are "lapped", and the other side is essentially independent. You'll know that you've accomplished your goal when the original stud starts to feel "loose", with only the nails holding it in place. If you have a reciprocating saw (Sawzall) you'll know the weight has been supported when the kerf of a cut through the stud no longer tries to pinch itself shut. Be wary of this method, though, because if there is still substantial pressure on the stud the kerf will begin to close as you pass the halfway point, which can result in kickback.
The shoring wall(s) should be at least as long as the section you're replacing. If it is indeed termite damage of the bottom plate, I would wonder how the wood could have supported any weight. I used to live in Wisconsin, where that type of construction (lath) was very common, as were full basements. Termites were unheard of, however.
Now I live in South Carolina.... Termite heaven. However, basements are very rare here... most homes are built on crawl spaces or slabs. If it is indeed termite damage and not just dry rot, I would highly recommend a thorough investigation of the rest of the house. The bottom plate in the center of a basement likely wouldn't have been a point of entry unless there is a crack in the floor nearby. If it was the point of entry, the termites would likely have began tunneling up the studs themselves as the colony grew and spread.
Look for "mud tunnels" or other very well defined tunnels chewed through the wood. If you find them, expect that they have also munched on other areas of the house. Hopefully it's just dry rot!
That wall is supporting ceiling joists so you can't simply remove it.
It's not carrying any roof or floor load as far as I can tell, so luckily no significant loads. If you want to clear span 13', use a 2 ply 1-3/4" x 9-1/2" 2.0E LVL. 1.55E LSL would be fine too, but that can be hard to find sometimes.
Typically remodelers try to put these types of beams up in the ceiling so they don't have to have a dropped beam with a soffit. They put the beam on top of the ceiling joists and use metal straps or strap hangers to carry each ceiling joist from above. The 2x6 ceiling joists will typically overlap a bit at the wall, so a 3" wide hanger every 16" can usually support both ceiling joists at the same time. It's a goofy way to do it, but each hanger would be carrying 200 lbs max so it could technically be done with just SDS screws but it's too much work to explain that to an inspector. They use hangers just to pass inspection easier.
Best Answer
In general, bolts are great fasteners, and unless the engineers did a particularly poor job designing the connection, you shouldn't have to worry about a properly assembled pull up bar. The issue comes when bolts start to loosen up and move.
If that bolt were to loosen up a bit, it would start to jiggle and move as you worked out. A pull-up bar will have 100s of pounds of force on it, so a loose connection will get looser and move more and more until the bolt strips.
You give the idea of "epoxy" on the threads, but there are actually "thread locker" compounds used on threaded connectors that are made for this. Any automotive parts store (and most home improvement stores) will sell thread locker. It typically comes in a "blue" color for removable connections and "red" for permanent, but you need to read the package. For this use, permanent should be fine, because even if you want to take down the pull-up bar and move it or sell it, you probably don't need to take it apart.
I wouldn't bother with a reenforcement strap. Make sure all the connections are tight and secure and you should be fine.