Knighthoods are the obvious first titles to hand out, due to firm rooting in English tradition and since a knighthood doesn't necessarily come with any lands or formal political power so there's little real cost to creating them. Beyond that, there's several directions things could go depending on how and why the US becomes a Monarchy.
On of Hamilton's early constitutional proposals had the President being elected for life and appointing State Governors who would serve at his pleasure. If you have a monarchy based on the Hamilton framework (with a King in place of the President), I could see "Governor" gradually acquiring noble connotations, especially if an expectation develops that Governors have tenure for life under good behavior (like judges IOTL) and that a retiring Governor's most suitable son is his logical successor.
If a monarchist constitution were to be drafted inspired in part by Cromwell's Protectorate, we could see the upper house of the Legislature filled by nomination for life by the King. "Senator" is the most likely title for these life peers, but the ATL framers could simply call them "Peers" or "Lords of Congress" (after the Scottish title "Lord of Parliament", equivalent to an English Baron), or they could find another historical inspiration other than the Romans.
If the monarch is a military strongman (e.g. if Washington had gone along with the
Newburgh Conspiracy), a likely move to consolidate power would be to assert control of the trans-Appalachian territories previously surrendered by the states to the Continental Congress and use Western land grants as a way of rewarding supporters and paying off important interests. These grants might come with titles, perhaps modeled off of the English peerage (Duke, Earl, Baron, etc), or perhaps modeled after Colonial titles ("Lord Proprietor" for the owner of a state-equivalent, and maybe the Cassique/Landgrave titles you alluded to for smaller or subsidiary titles, or even trying to obfuscate the creation of a new nobility by reusing titles used in republican state governments (Governor, Sheriff, etc).