I think it is possible if they work together. If they act as individual entities, then no they will not be able to. Prices go up when sellers overpower buyers (supply greater than demand) and vice-versa.

Which means, they need enough BTC to completely overpower retail and any opposing institution. If there is no-one buying at the lower levels, they could collaborate and sell/buy from each other to drop the prices to an acceptable level. The biggest problem they have is that BTC has a limited supply and it is also transparent. When they smash prices in the metals market, they dump so many contracts that it overpowers the buyers. This will cause holders to fold. However there are too many who plan to HODL BTC which means they are unable to fully drag down BTC prices unless they accept the risk of losing the BTC someone unwilling to sell.

There are still risks that crypto exchanges do transactions that bypass how much BTC they actual have (essentially scamming customers). Therefore if people were to withdraw all their BTC, they may not be able to pay our. This means there are too many factors to tell us for sure. It will depend on how much people are willing to pay to HODL, is the exchange ensuring they have enough BTC and whether enough of the financial institutions are willing to collaborate.