A Sevis registered school is likely to be more expensive than one which is not
Rubbish, there is absolutely no evidence to substantiate that claim.
In order for a school to be SEVIS approved it needs to be FAA Part 141 approved. This has to do with more stringent record keeping requirements.
Besides, under Part 141 you can do the IR right after the PPL, with Part 61 you need 50 hrs of PIC Cross country time before you can take your IFR checkride.
So per definition, a Part 141 PPL+ IR is 50 hrs. cheaper then a Part 61 PPL+ IR.
I'll leave it to you to add it up....
M1 visa and do the PPL training - then once you've finished and have 30 days to leave the country, get the IR done at a cheaper school
The school were you do your PPL is required to notify SEVIS that you training is completed. You are therefore no longer allowed to undertake any training since the training for which the visa was issued has been completed.
What you suggest will get you a visit from TSA and ICE (Immigration and Customs Enforcement) followedd by a free trip to the airport for your return home.
Keep in mind that immigration violations will get you on the "No Entry" list for up to 10 years.
Here's the good news:
You may change schools and keep the original visa if you decide to do so WITHIN 6 MONTHS of entering the USA.
You originally choose school A for your training and after 3 months you feel you may be better off by moving to school B.
School A is REQUIRED to transfer you to school B in the SEVIS system.
School B will issue the necessary paperwork, accept you in SEVIS and the training merrily carries on at school B.
Obviosly school B also needs to be SEVIS approved.