I am travelling from vadodra to Montreal. The following are my flight details
Flight 1: Jet airways 9W344
From BDQ (Vadodra) Departs at 20:40
To BOM (Mumbai) Arrives at 21:45
Arrives at Terminal 1B at Mumbai domestic airport.

After this I have an international flight with british airways
Flight British airways BA138
From BOM (Mumbai) Departs at 02:40
To LHR(London) Arives at 06:55

then,
Flight British Airways BA095
From LHR(London) Departs at 16:00
To YUL(Montreal) Arrives at 18:25

All the flights are on SINGLE ticket.

The checked -in baggage allowance on my first domestic flight with jet airways is 2 bags of 23 Kg. However, the baggage allowance for my british airways flight is 3 bags of 23 kg. (because I am travelling on student visa BA is giving me one extra bag).

I want to take the advantage of the 3rd bag that BA gives.
Now Jet airways provides the facility called through checkin where my luggage will be checked through from BDQ to YUL.
But I think I will carry 2 bags with me from BDQ to BOM and collect the 3rd bag from my relative at BOM before taking the BA flight

I am planning to inform jet Airways at BDQ (vadodra) not to do through check in of my luggage and check my luggage only till BOM. After arriving in BOM I shall manually collect my luggage walk out of domestic terminal and collect the 3rd bag from the relative then go to international terminal with 3 bags. At international terminal I shall check my luggage once again this time with BA.

My question is
Is through check in at Mumbai mandatory in my case?
If I want to avoid through check in at Mumbai and collect my luggage from Domestic terminal and check in again at international terminal will jet airways or British Airways object to it?

Will British Airways accept my luggage if I check them again in BOM or do I have to mandatorily use through check in?