Your question appeals to me in its clarity and relevance. I would love to give a similar clear answer, but as we stand in math and physics I believe that matters are not so clear-cut.
As pointed out in the comments above, no mathematics suffices to describe reality. But in physics we try nonetheless to describe parts of reality, and especially patterns that we believe to observe. For these patterns we also seek explanations in such a way that it helps us to delve deeper into the physical world. By delving deeper I mean that we gain more traction over the things we observe. For instance we now have 'access' to particle-physics processes, which enables us to study and observe yet more deeply the mysteries of the Planck-scale world.
Mathematics can also be seen as the science of patterns and (cor)relations. For physics, I believe it doesn't intrinsically matter which part of mathematics we use to describe a pattern, as long as the mathematical description fits our observations and helps us to gain more traction [I would like to say: 'helps us to gain more understanding', but history shows that what we in our time believe to be understanding will probably be labeled misunderstanding by generations to come...].
Now constructive mathematics offers a very beautiful and fruitful 'new' framework/perspective to look at the physical world. I personally believe that, in one very fundamental way, intuitionistic mathematics is more suited for physics than classical mathematics BUT...
in many other ways, classical mathematics offers the mind 'easy' approaches to explore new patterns and (cor)relations, precisely through not having to worry about feasibility, reality etc. This 'freedom to ignore boundaries' is very important I believe in ANY science, and I often find it distressing to see that unorthodox creativity and real originality have a hard time in our academic communities.
So, summarizing I would say that
a) there are no insurmountable obstacles to phrase our current physics in intuitionistic mathematics, but it would need hard work to do so
b) intuitionistic mathematics has a fundamental perspective to offer, both unifying and simplifying in nature, which I believe would go a long way in helping us delve deeper, so this work would be justified
c) it would be unwise to abandon classical mathematics for its ('unjustified') dreamlike abstractions, on the contrary we should embrace any mental dreams as long as they offer simplicity, beauty, mystery, etc. The drawback that I personally perceive with classical math in relation to physics is that the tacit acceptance of LEM and impredicativity often lead to mathematical machinery that is more complicated than necessary. And as I think to learn from history, simplicity and elegance are the gateway to progress in physics.
To answer your second subquestion: LEM plays an important role in so-called 'discrete' math systems in constructive mathematics. 'Discrete' then signifies that the equality of objects is decidable. For instance the algebraic numbers are a discrete subset of the complex numbers, and this gives a lot of constructive traction to 'constructivize' all sorts of classical theorems about the complex numbers. But in general of course LEM fails hopelessly. Certain choice principles are somewhat generally accepted in constructive mathematics, on the other hand most choice principles can be avoided by choosing careful definitions which incorporate the extra information necessary.
[Update 23 march to reflect the questions in the comments below:]
What I perceive as intuitionistic math's strong point for physics, is its insistence on potential rather than actual infinity.
This one strong point results in
a) a very elegant way of dealing with data from measurements, since in INT (intuitionistic math) one does not see real numbers as completed (infinite sequences of converging rationals), but as a sequence of shrinking intervals, which arises over time and which is never finished, but which can be pursued to an arbitrary precision.
b) a germane way of resolving one fundamental paradox in physics: is our world completely finite in all conceivable ways (which would be extremely hard to fathom) or is it fundamentally infinite in some way (which is equally hard to fathom...)
c) an interesting way of looking at (information) entropy since it implies that large numbers or large strings or whatever only arise over time...
d) a nice explanation why compactness might be a feature of the physical world (there is a sound math model called RUSS in which compactness fails, and I know of no good argument why classical math would describe the physical world better than RUSS...)
e) a nice explanation why physical processes are usually continuous, although this explanation becomes questionable at the Planck scale.
... well that should be enough for now.
In classical math, the main feature that I find very unpractical for physics is directly related to a) above. In classical mathematics we build the reals as equivalence classes of (say) Cauchy-sequences of rationals. But we never encounter equivalence classes when measuring/observing data. So it's very much like taking a lot of trouble to put your stuff in boxes (equivalence classes), and then each time you want to do something practical, you have to take everything out of these boxes again...
You may think that this is not so complicated, but for physics I disagree. When we look at the reals as arising from a pointfree setting, basically like Brouwer did and which is common in INT, then one gets a different intuitive picture of what is possible, for instance on the Planck scale, in terms of physical-world substrates underlying our measurements.
Basically, the whole idea of reals becomes simpler in the pointfree setting (well, if one takes care to simplify...) and to me this opens the mind for different avenues of exploration.