yes put them into the grow bag compost but in deep pots. it is better to sow them direct into the pots so you dont disturb the roots. you get better carrots this way. use the same method for parsnips too.

If you can get one of those little plastic 'blocking' machines, with the right peaty compost it is fairly easy to make compost 'blocks' These are good to sow into as you can simply plant the whole thing out once the carrots have germinated. An alternative is the peat pots where you do the same thing, but these seem to be ridiculously expensive. I've not tried newspaper pots, but that should work too in theory.

Why are you growing in seed trays?
I have never known any successful method of transplanting carrot seeds from them. Anything else for that matter.
There are many ways of growing carrots but all involve a good depth of compost/soil, unless you try those golf ball shaped ones which you may get away with in a grow bag.
Heavy clay soil unfortunately will never grow carrots successfully.

If you just want a few carrots to show your granddaughter, you'd be better off getting a container at least 10 inches deep, 10 inches diameter, sieve your grow bag and once sieved add an equal quantity of horticultural sand to it. Mix in a bit of growmore. Fill your container and then sow seeds of a very early maturing variety all over the surface, then cover with about 1/2 inch of compost, water well and cover the top with polythene until they germinate.
You must sieve any compost you use, preferably with 1/4" sieve.