I am looking for a solution to power a Raspberry Pi connected to a LCD screen 10 inch with three LiPo batteries 3.7V 5000mAh connected to two BMS 3S for charge and discharge. I designed the following circuit but I have doubts on its correctness. In case this is correct what is the expected battery life in hours?
Without the manual for the BMS it is hard to give an authoritative response, but I really don't think that you can connect these devices in parallel.
A BMS typically want "single access" to the batteries in question to control current and/or voltage. One BMS does not know how much current to add to the flow if there is another one which it has no control over.
It could work if there was a secondary communication of some sort between the boards, but it appears to lack such a feature.
In case this is correct which is the battery life in hours?
Your battery has MAX 63WH, your load is 7.5WH, so roughly 7.5 hours max runtime. You also have to factor-in efficiency losses, 10~15% and LiPo cycle health.
LiPo battery life cycle drops significantly when discharging them all the way down to 0% capacity. It's better to leave at least 30% in them, then recharge. Also they are not happy sitting full charged (4.20v per/cell), this also kills the life of the battery. Take these two factors in account for life cycle longevity.
Regarding the BMS question, the other posters are correct in "they/we" don't know for sure if they are OK with paralleling them, seek the answer to that or seek another single BMS that will do the job.