Why not move the seat of government entirely to the Lords?
We'd elect MPs, as now. They'd choose the PM, as now. But the PM would then receive an immediate earldom and move up to the Lords, forming his ministerial team there.
This would achieve the separation of the legislature and the executive that so many are talking about, leaving the Commons as a true legislature, holding the government to account and controlling the finances - and with no legislation at all required to achieve it (though doubtless Lords reform would follow swiftly).