Ah - but if we exclude the electromagnetic force (which I think we can)
then the force that could constrain the Higgs particle would be either
the weak or strong nuclear force.

And yes they are countless billion times stronger than gravity - but
only at very short range. Once you are more than a few nuclear
dimensions apart the nuclear forces quickly drop to near zero.

Only gravity and electromagnetic forces operate over long ranges.

So if the nuclear forces were constraining the Higgs particle there must
be an awful lot of them very close together. On the other hand, if
there were slightly less of them, gravity should take over and we would
get bunching.