Intuitively, an independent airline or affiliate would be able to apply for more slots at Heathrow despite flying for that airline - such as BMed is able to do - therefore increasing slot penetration at Heathrow and opportunities on more profitable routes - or lowering the cost base on existing ones.
Whether BA has the ability to acquire a stake is another matter - though its debt burden is now low enough to allow greater borrowing to fund acquisitions and purchases, it may be the case that with an impending reduction in capital when pension liabilities are brought on balance sheet that an investment of that kind might be more of a burden than not.
Ideally an acquisition should be able to make use of economies of scale - what if any could be made by such appear low as the workforces would ultimately demand integration and reversal of those cost benefits. If they acquire just a stake, they cannot control it to do as you suggest; if they acquite a majority stake and hence control, the benefits of slot acquisition are negated. A transferral of routes down to a lower cost base at the same flying base would also negate many union agreements.
So maybe - if there was a detailed analysis performed, but probably not.