First do the written before August. Then you may be able to make it work. With a couple "ifs." If you you get the 61.75 "based on" certificate before you do the checkride for the ATP and if you sit the regular FAA IR written you will then have an FAA private with an IR. If you can show the training for your type ratings was conducted by a FAA instructor the types will transfer to your FAA private certificate. Then when you take the ATP the types will transfer to the ATP without a checkride.
There are several places where you can do the 737 rating for ~ $8000. The 777 is going to be higher, not as many places do the training so no competition.