Hi Peter,
I’ve never heard of anyone needing an invitation to come to Ecuador to apply for the 12-IX visa. I just reviewed the requirements on the Ecuadorian Embassy’s website and there is no mention of needing invitations. You can find the requirements here.
As for the bus – I don’t think you’ll be able to purchase bus tickets in advance. I would speak with someone at your nearest Ecuadorian embassy and ask what they suggest. I do know people who were able to book airline tickets through certain websites that allow a refund if you cancel within a day or two. Once they booked, they printed their flight confirmation, then cancelled the flight and received a refund. It’s not the best method, but one that might work if you can’t find any other options.