There is a more general question lurking in the background, which is what do critics of logical foundations generally think about applications?

Historically, intuitionism is not the only foundational controversy. Earlier, there were critics of the logical foundations of calculus (recall Berkeley's "ghosts of departed quantities"), and today there are unresolved mathematical difficulties in quantum field theory. The pattern is usually the same. The practitioners, based on intuition and experience, know what they have to do to make sure that "bridges don't fall down," while the critics point out that the practitioners have failed to articulate clearly what the ground rules are. In particular, the critics can sometimes construct calculations that appear to avoid all explicitly forbidden operations, yet yield the wrong answer.

To determine what a specific person (e.g., Brouwer) thought, one obviously needs to examine what that particular person said on the subject. But in general, there will be a range of opinions. Some, as you say, will believe that the scientific/engineering theories must be fundamentally correct even though we haven't hammered out all the logical details yet, while others of a more alarmist bent may worry that people are trusting the theories too much, to the point where a "bridge will fall down."

Obviously, in the real world, bridges do sometimes fall down, but there are many reasons for this, most of which have nothing to do with inadequate logical hygiene. I would be curious to know if there are any examples of an actual engineering disaster with the following features:

The disaster can be traced to a calculation that the engineers mistakenly trusted (and not because there was an inadvertent error or bug).

If one were to apply "more rigorous reasoning" then the calculation would have come out differently and the disaster would have been averted.

I can imagine that there might be modern examples where the results of some kind of numerical simulation are trusted, but where theoreticians can show that the numerical results do not accurately reflect the behavior of the equations. (If the equations themselves are an inadequate model of physical reality, then that is a different matter, which I'm not concerned with here.) But in general, I think such situations are rare, because theoretical predictions are usually tested experimentally before an actual engineering project that might endanger people's lives is carried out. If I am right about this then most fears of "bridges falling down" because of lack of logical rigor are overblown.

Et ceteranonet cetra. In fact, in this case,et ceteri, as the poor guys are not objects. However, one usually abbreviates&c.$\endgroup$