I often like to pay the house fee by showing the amount of cash I have on me that I’m “willing to spend” (this amount is still less than the total I’m willing to spend, but shows them that I have more to give them - I.e if the hf is .6 for 1 hour, I’ll take out 1.2 and give them .6 while having a separate 1.0 that I haven’t shown, and sometimes, depending on the establishment, won’t have to even use).
I’ll give the hf and then if/when tip negotiations take place, depending on what’s on the menu, I’ll offer the additional .6 that they think is all that I have (standard at most places where the menu is limited), the additional 1.0 (at places where the menu has a bit more to offer), or the full allotment if I’m going somewhere with an all-you-can-eat offering.
At the end of the day there’s no real protocol and varies place to place, I just like to not have to halt the action unnecessarily.