Coq uses type classes to infer h-levels in some simple cases automatically (see Sect. 3.2). Agda does not have an automatic way to manage or infer h-levels. I agree that one has to do quite a bit of bookkeeping to carry around proofs that something is of a certain h-level; and proving that newly introduced data types preserve a certain h-level requires quite lengthy encode-decode proofs (see for instance this proof that lists preserve the h-level of the carrier)
It's important to keep in mind that current formalizations of HoTT are still quite prototypical and based on proof assistants which had been devised before the idea of h-levels came to type theory. Future theorem provers will likely give more support to work with h-levels, possibly by introducing h-levels judgmentally like Arend does (thereby giving an orthogonal stratification of types to universe levels).
Automating the bookkeeping should be an engineering task, deriving automated proofs about the h-level of a type or filling cubes in Cubical Agda is in general undecidable. I'm not aware of any work in this direction, but there are lots of interesting things to be done. Similarly to, e.g., termination checking, there are probably lots of things that can be done automatically even if the problem can't be solved in general.