There are two premises in this argument.
The first premise is that the trade war undermines global economic institutions. Three problems stand out with this claim:
First, this link is tenuous at best: a large number of trade disputes regularly occur around the world that don’t flow through global economic institutions, so it’s difficult to maintain the claim that this particular dispute undermines their credibility in any unique way.
Second, this claim misattributes the source of the damage to global economic institutions. Even if it is true that the Trump Administration has attacked global economic institutions, the trade war did not cause those hostilities – it is the effect of a pre-existing antagonism with the global economic order. The trade war is the effect, rather than the cause, of this antipathy. So it is difficult to attribute the broader attack on global institutions to the trade war specifically.
Third, this claim conflates the ‘global international order’ with the ‘US-led global international order’. If it is true, as the argument claims, that a rules-based international order is beneficial for countries in general, then they will continue abiding by a rules-based order – just without the US at the helm. We see this already with free trade agreements like the Trans-Pacific Partnership proceeding, absent the US. The rules-based order will survive, with a potential reorganization without the US at its center.
The second premise is that global economic institutions are good for the world. If it is true, as President Trump claims, that these institutions stack the deck unfairly, then this premise is not obviously true, though it is worth noting that the Trump Administration has not supplied strong evidence to that conclusion.