The cash – from the Government's Cultural Recovery Fund – comes on top of £2.56m the museum received in October to help keep it going during the pandemic.
It will fund its Forging Ahead project, which will see a new development at the site off Tipton Road, Dudley, focusing on the 1940s to 1960s, including a new visitor centre and car park.
Meanwhile three independent cinemas in the region have been handed grant support of more than £500,000.
Andrew Lovett, chief executive of BCLM, said the project was "more important than ever" to the museum's future in light of the Covid crisis.
He said: "As the biggest single development in our 43-year history, Forging Ahead gives us the momentum to thrive once again for our community following the unprecedented difficulties presented by the pandemic.
"Forging Ahead is not just about exploring the past, it’s also about making a positive difference to the future of this community too."
The project is expected to help create 140 new jobs and contribute £102.5m to the economy by 2032.
BCLM has been closed to visitors during lockdown and is currently operating as a mass vaccination centre.
Dudley North MP Marco Longhi, said: "This huge level of investment reflects the priorities for the BCLM and its future.
"This really is a significant investment for a world-renowned site, which we are lucky to have on our doorstep.
"I believe the Forging Ahead project shows the importance of building on the historic success of the museum and forging an even better future."
The museum also received £1.175m from Arts Council England's Emergency Response Fund in July.
The British Film Institute has awarded £5m to 33 cinemas, including Reel Cinemas in Halesowen and Bridgnorth (The Majestic), which have each received £115,385 and £292,836 for The Light Cinema in Walsall.
KC Suri, head of Reel Cinemas, said: "This vital funding will enable us to protect job security and to manage our business sustainability at this challenging time."
Keith Pullinger, founder of The Light Cinemas, said: "Covid-19 has devastated our finances and this grant is essential to protect our future and ensure our survival.
"There is still a lot of work to do, but we now have a platform to start rebuilding our business from."
The awards were announced as the final allocations from the first round of the Government's £1.57bn Cultural Recovery Fund.
Culture Secretary Oliver Dowden said the fund was "helping to save the places people can’t wait to get back to when it is safe to do so".