Not unless you've invented a new form of mathematics to go along with it, it won't.
The "long term saving" is going to be, at best, the current amount written off per year plus inflation plus any any % increase in the number of MPs. The costs involved in your approach would continue to be vast...
They're disputed amounts, not proven false expenses. Vast, vast difference.
There is a good reason these have been written off - they're not doing it to be nice, they're doing it to save money.
So you've no workable suggestion then.
I imagine you'd be spitting fire if your employer refused to pay an expense that you claimed was legitimate as they were going to "get around to it eventually", seeing as you've got yourself so worked up over this case of the media making a huge issue of...
Who do you expect them to dispute them to? A person or persons are required for this.
You are not going to find a cost-neutral way or even a minorly loss-making way to get them back. There is a damn good reason they've been written off.
These are all disputed claims. Disputes require someone to work through them. That isn't free.
Its £3 per MP per year. That isn't enough to consider "abuse" and they've just given themselves rather a lot higher payrise than that, if you want something to rally against.
Its 26 MPs out of 650; the amounts involved are tiny compared to the overall amount paid out.
Would you rather then spent more taxpayers money getting it back? It'd be significantly more, you aren't going to get a staff member for £2,100 a year.
If you can rationally explain how that's a workable comparison, go ahead. (hint: its not, its whataboutery)
If these are even 1% of their expenses to begin with I'd be astounded also. Even the one over £300 is probably a fraction of 1% of their expenses for a year, unless their constituency...