Heaters are popular here in NZ... and unfortunately to fight safely means to avoid some really nice martial techniques
. However I quite like using a polearm against a shield user, doing a shot to the head, then a shot to the leg. It either works or they get very tired arms, and then they make a mistake
. The flip side is that a polearm user is vulnerable to being bound up by the shield. But this may not apply in your sparring system, as rules change the techniques that work. Tournament is not war. Having said that Sir William Marshall was talented in both, so maybe there is some crossover?
Thrusting past a shield, hooking it on the retraction and then thrusting again on a new line can work well too. Works even better with a buddy.
Hopefully someday someone discovers medieval manuscript buried somewhere. It'd be nice to know what was taught, and if our speculations are anywhere near the mark.