BRP Expiry date

[Spiderman2010]

Hi all,

I applied for Tier 2 visa to work in the UK at the end of last year. My COS was issued with a wrong start date (11/2018 to 11/2019). I informed the HR and they added a note on the COS that the post start date is 1/2019. But they didn’t mention the end date.

Unfortunately my BRP was issued based on the old dates and it’s valid only until 11/2019. My contract is for one year (from 1/2019 to 1/2020) which means that my BRP will expire 2 months before the end of my one year contract.

I am planning to change jobs and I will apply for a new visa sponsored by a new employer. My contract notice period is 3 months which means if I resign now then I will need to work with my current employer until December (about one month after the expiry of my current BRP).

My questions are:
Can I apply for a new BRP sponsored by a new employer and continue working with my current employer for one month after the expiry of my current BRP then join the new job?
Is my contract expiring by the expiry of my BRP or do I still need to give a 3 month notice period?

According to the HR I can’t work with them with a new visa from different employer and I need to apply for a visa extension which means that I will apply and pay visa costs for 2 times in order to cover the one month gap between the expiry of my BRP and the joining date of the new job which is also not practical because of visa processing time. Any other suggestions/ solutions?