I think that's the only way to do it at home. Maybe Kinkos or some online service could do it as one big sheet, then fold the envelope yourself. Like you said, it'd use a load of ink to do one, let alone all 4 of them (with inserts). Either way, it is definitely a printed font as the letters are all the same. I assumed they did it the way I mentioned because some of the "E" letters don't have the little hole in the top and others do. I assumed that was ink bleeding or something.
Also, I was looking at the reference images today. I was going to look into buying all of the stamps that I could and then I realized that they're not even stamps. I think the Royal Mail Queen head stamps are but most of the others look like printed squares, which were then cut with those special craft scissors. Some of them even look like the just have messy, rushed, notched triangles in them instead of the circular stamp edge.
You can definitely buy stamp edge scissors though so that would save a lot of money just printing/cutting the stamps instead of buying them all.