Unless she has a history of using a lot of visa-exempt entries she should be fine doing a border bounce. Rather than flying she could use an agency to take her to a land border. The agencies guarantee re-entry without problems.
That's a multi entry tourist visa METV. It last for 6 months but if you time your border bounces you can get nearly 9 months. On your last entry there is nothing to stop you leaving and returning then getting a visa-exempt entry. They don't do a one year visa now.
You'll both need re-entry permits to keep your permission of stay valid. The baby will get a free 60-day visa-exempt entry. Children under 15 do not get fined for overstays should you have any problems in the future.
Just book in a cheap hotel for one night before going to do your extension. You only have to take a printed copy of TM30 if you stay in private accommodation. The hotel will do your TM30 online so just take proof of your booking.
Tourists can work online if their work doesn't involve any business in Thailand. As a UK citizen, you will be stamped in for a free 60-day visa-exempt stay.