Tea bag disposal plate name

Small Sauce Plate.

If you look up 'small sauce plate' or 'small dipping plate' online, and click 'images', you'll see lots of these kinds of tiny dishes, so I think if you ask for one of those, the waiter will understand what you mean:

https://www.amazon.com/small-dipping-plates/s?k=small+dipping+plates

I don't think there is really a name for that small-plate thingy as a specific 'tea-bag plate' per se. Instead of asking 'may I have a small plate for my tea-bag, please' which as you say, is quite annoying, I suggest you ask for 'a small sauce plate'.

An Alternate Answer.

Teapot.

A teapot is really the correct place for the teabag. How about ordering tea by the pot?

The teabag stays in the pot - and you can request extra hot water to top up, giving you even more tea, for free!

I'm English, and a lover of tea, the teapot avoids the inelegance of this 'tea-bag problem'.

I get your little plate idea, though as I say, tea is better in a pot - however I do have a tiny, shell-shaped, white porcelain plate, only about 4-5 cm across, and quite flat. Could something like that fit in your bag? Then at the end, the used tea-bag goes in the tea-cup and the shell plate is dried with a tissue and goes back in your bag. It is really, a small sauce plate. And that's what we use it for, in my cafe.

Tea-time, for English people is a peaceful little moment around the tea-pot, where we can connect with another and share what's really important to us. So that's another lovely thing about the tea-pot - to share your tea-time with a friend!

[Lakeland White Porcelain] Tea Bag Rest "Where do you put your tea bag after you’ve fished it out of your mug, or taken it out of your teapot so your brew doesn’t stew? On our pretty little White Porcelain Tea Bag Rest, of course".

Brought to you straight from the UK. [ha ha]

Also, for spoons and other utensils when cooking:

Here is a spoon rest (which I am going to buy as the one I had for 30 years broke.) from William Sonoma:

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As has been pointed out by FumbleFingers, in a comment, some merchants who sell such things refer to them as teabag holders.

While that term is adequate in the context of a website on which it is accompanied by pictures and descriptions, it is debatable whether it would be readily understood to have this meaning in the absence of such clues; upon encountering it in isolation, some people could think that it stands for something that holds teabags before use.

A leading global online retailer, nominally based in Brazil, has the following that meet your requirements, and perhaps more that I haven't yet seen.

Tea Bag ...

• Dish
• Tidy
• Coaster
• Holder
• Rest