I always worked on the basis that the lieutenant could spend the "basic" regency of the action (as you've told them to do the action and everybody knows it) but not spend additional RP to boost the chance of success as that's above the basic effort - it lets them do a lot more actions but not deplete the treasury much although from one perspective it doesn't make much sense.

There is a magic item (gloves of regency?) that let you store 12 or so RP that a lieutenant could then spend from recollection.

In practice though the bureaucracy isn't powerful enough to let a lieutenant do a lot, the "king" is still, well, "king" and so vital for most things. Which is good in one way as an over-powerful lieutenant could easily become a great captain or risk undermining the ruler (for an example of a lieutenant getting overly powerful in they're own right look at Muden)