I am assuming you are of the school that holds that Jefferson had no authority under the Constitution do make that deal.
If I am mistaken, then disregard the following."
No, I am of the school that holds that "strict constructionist" Jefferson would have opposed the Louisiana Purchase, on Constitutional grounds, if President Adams proposed it.
And we know that for certain because Jefferson's erstwhile ally in strict construction, Tertium Quid Virginian John Randolp, the Old Republican, he did just that, in alliance with many Federalists, on Constitutional grounds.
Randolph's constitutional objections were only defeated by two votes -- 59-57,
Further, the excuse that you made for Jefferson -- the Federal treaty-making authority -- is not the one Jefferson himself made.
Jefferson himself only made the argument that Louisiana Purchase was one h*ll of a good idea and good deal.
Jefferson said nothing about its constitutionality.
Jefferson's Secretary of Treasury, Albert Gallatin, is the one who used the Federal government's treaty making authority to justify the Louisiana Purchase.
And doubtless, today such arguments would be accepted immediately, but it worthy of note that at the time no true Strict Constructionist, such as Old Republican John Randolph, were buying it.
My point in all this is: like all Democrats since, Jefferson was a total fraud and hypocrite regarding "strict construction" against Federalists.
"Strict Construction" for Jefferson was simply a weapon to be used against Federalists, just as today Democrats still weaponize "strict construction" whenever they can, against Republicans -- not because Democrats believe a word of it, but because they know Republicans do believe and care about Founders' original intent.