MEng vs MSc, US & UK

I'm currently studying an integrated masters MEng Mechanical Engineering in the UK and moving into the 3rd year of my 4-year degree. I could choose to downgrade my degree to a BEng and graduate this year, or continue for another year and graduate with an MEng degree.

I would prefer to work in the US if possible.

I've seen online that MSc is the standard in the US and was wondering if I should aim to graduate a year early and pursue an MSc in the US (or in the UK if that would make a difference), or if the MEng course will be relevant/competitive enough to get a job in the US.

Does anyone here have any experience or knowledge that could help? It'd be greatly appreciated!