Scroll down the page and you will see a story from someone who got denied boarding exactly like this. Immigration supposedly share info with airlines and they will deny you boarding if your name comes up.
It needs to be transferred into your account from abroad and be in there for two or three months. Can't remember how long this second. But you don't need this for the first 90 day single entry. You can go and get a single entry from Savannakhet easily and apply for the year visa when you are back in the country. I'd recommend Savannakhet as they are fairly relaxed about everything there and you don't need an appointment. Fly from Don Muang to Nakhon Phanom and then get a minivan to Mukdahan border crossing. Then across the bridge to Laos. Very easy process.
I've done the whole process for as little as 7-8k before. But if you want to fly and enjoy yourself a bit when your there about 15-20k. 15 is well doable if you watch your cash.
for the non immigrant 90 day single entry it should be fairly easy to do it all yourself for about 20k. It'll be way better than a tourist visa or visa exempt.
You should get yourself a non immigrant O visa on the basis of having a Thai child. It'll give you 90 days and then you can extend for a year if you have the funds. If not a year then a 60 days extension at your local immigration office. If you're name is on the birth certificate it should be fairly straightforward. Best place to do it is Savannakhet Laos. Catch a flight to Nakhon Phanom from Don Muang. Then a mini van to the Mukdahan boarder crossing. The way the flights work out usually means you have to stay over two nights. This is your best option for now because your rights as a parent are more than just a tourist. Oh and you can also get your work permit on the visa if you manage to extend it to a year. It should be possible with either 400,000 baht deposit into your acc from abroad or a 40,000 baht income from abroad. Having this should help reduce the headache of getting a work visa and make the process of getting a work permit a bit easier.