I probs wouldn't ask this because I'm also socially awkward and shy and would be really embarrassed if they said no and would never be able to go in there again, BUT....
I think given the context you weren't unreasonable to think of asking.
You're a regular customer, they know you use them quite often and always have the money to pay, you obviously like their takeaway because you keep going back, so if you were to take the food and never pay you wouldn't ever be able to go there again, so would have lost a good takeaway place. I think it would be good customer service from them if they were to weigh all of that up and decide that its highly unlikely you'll fail to pay, and even if you did they could just ban you and they've only lost a meal. The most likely outcome is that you do come back and pay, and continue to use them, and they have delivered you good customer service so you're likely to use them even more, recommend to others, leave them a tip etc.
Tesco/Mcdonalds is different to a local takeaway. If Tesco allowed someone to walk away with goods without paying it would be all over the internet and everyone would try it. I doubt people will travel from miles around to visit the Chinese takeaway that let's you take food home if you 'forget' your card. Also they're independent so it's at their discretion to decide who they want to do favours for i.e regular, loyal customers.
Also if someone was allowed to do this at Tesco, they could easily not come back to pay and just keep going to different Tesco stores doing the same thing. In this context the OP would lose a favoured takeaway place if she were to not pay, and I think its reasonable for OP to think the owner of the takeaway would conclude that its unlikely she'll not sacrifice her regular takeaway just for one free meal. So, bit silly to compare the situation to a Tesco or Mcdonalds.
Of course, the Chinese takeaway owner would not be unreasonable to refuse your request, but I don't think you were unreasonable to ask. If you don't ask you don't get and all that.
I sympathise with your DH though as I personally I wouldn't have wanted to ask them either and would have used credit card, or picked up cold food and reheated it. He's unreasonable for going into a 'strop' though. and the comparison to Tesco doesn't hold up for me.
Not a big deal though - I don't think there's a right or wrong as far as wanting to ask goes. Its just differing personalities/opinions. Agree to disagree. And tell him to remember his card next time!