Let's make all MPs and their private offices into small companies, subject to all of the same rules as contractors. Pay them a reasonable daily rate for days they work, and make them subject to their employers' (constituents) view on how they perform, with a month by month review, and subject to immediate termination if external factors change? And if they breach the terms of their contract (the manifesto on which they were elected), then they're out. There's a news story today about them employing relatives in their private office - this gets rid of that problem too, they are free to do so if those people are the right person for the job, and all of those costs (and all of the cost of their offices, their travel, housing etc) would all be covered by exactly the same rules as limited company contractors.
Alternatively, make them all permanent employees - with one "normal" place of work either in constituency or in Parliament, no expenses while they are there or to travel there, but they can claim cheapest travel option to the other with an overnight allowance of £30 plus accommodation either from their own pocket or at a budget hotel within reasonable distance. No private office (normal employees don't have one) but a shared resource pool, no second home as an expense, no bonus at the end of a contract, and certainly no final salary pension.
After all, if contractors get the same benefits as others (which we don't) and should pay the same tax, or alternatively if the employee tax system is entirely fair and reasonable, then MPs couldn't possibly be so hypocritical as to give themselves benefits others dont get, and at the same time oversee a clampdown on others, could they?