I just find it amusing that it's the 'make 'em eat it' bunch who assume a child would be silly about food and refuse to eat things they've previously liked and turn their noses up at other peoples' food. Mine certainly wouldn't.
Being at the table is important: table manners are important. Discreetly leaving something you're not keen on is much better than making a to-do about it, and everyone, adults and children alike, should be free to do this. If the issue was the mucking about, deal with that, and separately from what the child does or does not want to eat. I suspect the fake coughing and all that was to do with the anticipation she'd be made to 'try a bit' of something she did not want.
My children have often come home from other peoples' houses and said the fajitas weren't like at home, or the bolognese wasn't as nice - but always ate it up, often better than the children whose house they were visiting. But in your own home, you should be free not to eat something you don't want.
I'm not indulgent about it - it used to drive me bonkers when dd was small and MIL would see she wasn't eating much of her first course and start leaping up saying 'would she like something else? Would she like a yoghurt? What about some smarties, dd?'. My policy is ignore what she is or isn't eating as I would an adult - the only thing I wanted her to learn at the table was how to behave at the table, which is that pudding comes when it comes, you wait for everyone to finish, you sit and behave and join in the conversation. When your pudding comes, if there is one, the same applies. I would only offer a sandwich if I'd said before 'look, I know this is a new food and if you really don't like it, I'll make you a sandwich, but let's give it a go, ok?'
Oh and all of you who would 'only' offer a slice of bread and butter - as a picky child, a slice of bread and butter would have been much preferable to whatever else was on offer, and I'd've bitten your arm off for the chance to sit and eat bread instead, so you might want to rethink that as a punishment! Gruel, maybe?