I always saw it that Lambert, after devising the Instrument of Government, developed a more entrenched republicanism; he wanted more balances and checks on executive power. This might be why he toyed with the idea of ousting Cromwell altogether.

Like many other experienced officers of the Army, he could conceive of something that transcended kingship after his grueling battles with the royalist enclaves in the North. Maybe I'm wrong, but that's how I see it.