My Dyson DC07 isn't sucking the way it used to, despite replacing the
lifetime filter.

In Jim Royle voice "Lifetime filter, my arse!".

If it's anything like the DC08 there is a recessed small clip which allows
the cover to lift up.
To test for blocked filters etc you can run it with them removed and see if
suction improves.
You may have something stuck in one of the tubes somewhere restricting the
flow.
I found a pressure-washer to be a very effective tool for cleaning out the
dust collection bowl and plastic pipework into the cyclones above it.
(Removed from cleaner of course!)

Alan.