There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when thinking about metamathematics? Of course, there are many different "levels" of intuitionism and constructivism, but it seems strange to me that a constructionist would rely on results like the Löwenheim–Skolem theorem, when it uses LEM at the metamathematical level. Isn't assuming LEM, even if at a metamathematical level, begging the question for an intuitionist? Are there philosophical reasons to believe that it may be more or less acceptable to use LEM at the metamathematical level from a constructive standpoint?
Furthermore, is there a computational interpretation of metamathematics in general? Its just a hunch, but it seems to me like metamathematics, and using LEM at the metamathematical level might be related to oracles in some way, but I have no sources for this.