All depends if taking the strainer off and disassembling is an option.
Sounds if the lid is frozen, the whole thing could use a tune up.
If apart, lots of the suggestions will probably work.
I have found I have broken less things, such as bolts by using a very long lever arm with tiny in cements of force rather than a shorter one with more muscle. Tapping and PB Blaster to the underside with alternating heat or cold applied will probable get it.
Just the hassle of taking it out and apart is the ugly issue.
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