This sounds pretty hard to deal with. It seems like she almost can't swallow or spit it out rather than her doing it specifically to be silly or play games etc. The reason I say this is because I get the impression that she seems so upset by it. Was she actually trying to put the ice-cream in her mouth while she still had food in it and getting upset because she couldn't?
If so she is obviously causing herself some distress.
Maybe she is genuinely having some sort of problem with actually swallowing which she probably doesn't fully undertand herself.
I'd be tempted to try the ignoring it a while longer, however if you sense that she really is upset, she may be asking for some sort of attention and given the stress this is causing everyone a different tack might be in order
You could try taking her on your lap when she first does this and giving her a hug. Say something like 'It looks to me like you are getting upset with that in your mouth, can you spit it out or swallow it?'
If she shakes her head or ignores you, maybe say something like ' Well that's OK we all have things we find difficult sometimes but mummy and daddy are here to help you' Then maybe ask her if she wants to open her mouth so you can clear out the food for her. If she doesn't want to do that maybe say 'just come and ask for help when you are ready' and try more cuddling.
This might go on for a while obviously, in which case you can keep saying through the evening -'when you are ready to swallow or spit that out or to let mummy/daddy take it out let us know'
Whe it comes to teethbrushing time, you could be firm and say she needs to open her mouth so she can brush her teeth at which point you or your dh could remove the food for her.
She may well cry at some point but at least then you can take the food out while her mouth is open and then you could maybe take that opportunity to talk about what happened at that point when the incident is still fresh for her. You may not get much sense out of her but she may feel a bit better.
In the end even if it doesn't work you may feel less irritated by the whole thing if you see it as something she geniuinely is struggling with rather than something she is doing to annoy.
I suppose this is pretty conflicting advice but I believe ignoring can work for mild silly ehaviour but if it is something that is getting entrenched and the child is obviously distressed maybe it is time to pay the attention the child is obviously asking for.
I'd suggest trying to make her laugh but I'd be worried about choking if she laughs too much with her mouth full.