Expression for indirectly asking someone to do something

I think that either hint or insinuate works quite well in this situation, and that you don't need to look for something else. Any different word is going to sound awkward and not be as natural a fit.

For instance, you could use euphemism:

[Merriam-Webster]

: the substitution of an agreeable or inoffensive expression for one that may offend or suggest something unpleasant

As in:

The waiter used euphemisms to suggest you finish your meal ASAP.

But that's not as natural or meaningful as the simpler hint or insinuate.

That aside, the only idiom I can think that applies (and it's the opposite, although the situation could be adopted to include it anyway) is beat around the bush:

[Merriam-Webster]

beat about the bush or beat around the bush

: to fail or refuse to come to the point in discourse
// Stop beating around the bush and tell me what you want.