The act of not funding the STL is purely symbolic, and doesn't make sense if Mikati and his government aren't officially in confrontation with the STL. As everyone knows his position is not the same of HA's position which is more confrontational. If it is a deliberate ploy, then fund it by all means. If Mikati wouldn't have it any other way, as he is not politically capable to challege Hariri for the premiership and at the same time enter in a confrontation with the STL, then it also makes sense to fund it for now until Mikati is politically strong enough to enter in a confrontation publically. Funding or not funding the STL is dictated by Mikati government's overall dealing with the STL.
Well HA wanted Hariri specifically to cut the fund for STL and nobody else.
And they probably got a good reason why they wanted only Hariri to take such step..