Multiple reasons. First Gustav III was a king who believed in the idea of 'Enlightened Absolutism'. He'd instituted absolute monarchy (one of two quite short periods of that in Swedish history) in a coup d'etat, inspired strongly by Louis's absolutism, and Gustav in his coup strongly curtailed the power of the Swedish parliament. Gustav wanted to be hip to his age ('enlightened') but firmly believed in his - and other kings' - divine right to rule. (and indeed a huge stickler for rules and traditional formality in general, to the extent that other rulers like his cousin Catherine the Great found him annoying to deal with) His 'enlightened' ideas were - for instance - things like abolishing torture, but not popular rule.

Second, Gustav (as many others in Europe at that time) was a huge Francophile and admirer of France and the French monarchy, Versailles and all that. His court used French a lot more the Swedish language. (and the late 18th century is the one period in Swedish language history when loanwords from French dominated)

Third, per the above he didn't have so much to gain from aiding Louis as he stood to lose by not doing so. Louis's downfall started earlier but Gustav III still ended up with the dubious honor of being murdered first.