Usually, you'll have to run a centrifugal compressor, with a suction surge drum, to enable this strategy. If the loads are variable, you'll need a variable speed driver, such as a steam turbine. There are the usual mechanical reliability issues with rotating machinery. Also, you'll need to think carefully about anti-surge controls to protect the compressor against very low inlet flows.
The economic justification would require a careful analysis of the net present worth of the installed hardware versus the economic value of the heat recovery, after deducting the operating costs of the compression system.