For the compulsory insurance - you can get this from 7-11 (of Lotus's's's's or Big C) for 100 THB. The insurance certificate has a tear off section along the bottom that is used by the DLT office for tax renewal.
With a bike - you don't need to queue up at the drive through lane, just drop into the DLT office, take a queue ticket for the tax (not the one for driving licence renewal, which can be a long queue), then hand over your insurance document and green book plus 100 THB for a small bike (107 THB for my 300), and 5 minutes later you'll get your tax sticker.
I went to the DLT office on Sukhumvit Road, tax renewal is on the upper floor, it took less than 10 minutes in total.
Also possible to do the whole process in 7-11, but you need a Thai ID card, and you have to wait for the tax sticker, so better to go to the office. All signage was in Thai, but it's easy to figure out the process, and staff were helpful.