Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented groups with quadratic Dehn functions?
Yes, the highest known Dehn function of a subgroup of a finitely presented group with quadratic Dehn function is exponential. The example was found by Yves Cornulier and Romain Tessera in Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups. Confluentes Math. 2 (2010), no. 4, 431–443.