Some "schools" of teaching start both children and beginning adults with the Thumb-Under-Frog hold and just call it a beginner hold.  Graduating to the "professional" (more standard Franco Belgian) is something they do some months later, as the stuent is comfortable with it and *needs* it for techniques that need a somewhat more "agile" bow.  

So I don't think that you are going to do yourself some huge artistic damage by using it for at least a while.  But in maybe, I dunno, 6 months or so?   The thumb may be sttrengthened and limbered enough from practice and playing that the more standard Franco Belgian hold that a lot of people think of as the only "proper" way of holding the bow may no longer be any problem for you. 

Or it may not, since the discomfort may be due to the old injury.  I'm not a doctor or physical therapist, and I don't know all the details and xrays and all that of your thumb's situation.  Neither does your violin teacher, unless she has had medical training on such things (which would not usually be included automatically with any level of violin playing or teaching, so far as I know). 

So far as tensing up your thumb, that isn't good in TUF hold, either.  Put simply, the tighter you hold the bow, the less flexible youyr bow hand is, and that will not help your playing.  Even using TUF, try to keep the thumb gentle and curved.  Curved forward, that is, not backwards.  Gripping hard enough to bend the thumb backwards, you're locking the thumb joint, and it limits the range of movement that you may need later. 

So my thought would be maybe you can keep with the TUF for now, since the FB (Franco-Belgian) hold is causing you pain.   That may or may not get better after a few months, no telling with your injury.  But relax the thumb and work on keeping it gently curved anyway.  In any hold I can think of, a tense thumb isn't going to make playing any easier or better.  The TUF *may* be a bit of an obstacle for some techniques you may want to learn at the level you want to be able to do them at.. Later.  But for beginning, it isn't such a bad thing, and it is the standard beginners' hold with some teachers.  

Maybe you can "grow out of it" later.  If not, I have seen some fiddlers who certainly seemed to play quite well with a TUF hold.  I wouldn't worry too much about using TUF at the moment, if it saves you some pain.  Pain is not something to "push through" in a case like this (or most cases in music).. It is a warning from the body that there is a problem.