Is there a spell that lowers the spell resistance or increase the "overcome spell resistance" of the caster?
I'm not aware of any spell, prayer, or invocation that lowers opponent's Spell Resistance protection, nor any magical affect that increases one's chance to beat an opponent's Spell Resistance protection, but I do know of two feats that should fill the bill: Spell Penetration and Greater Spell Penetration; each adds to one's chance to break through opponent's Spell Resistance. And they are both core feats.
In 3.5 there was assay spell resistance, in pf I do not know any.
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Piercing spell metamagic feat:
Piercing Spell (Metamagic)
Your studies have helped you develop methods to overcome spell resistance.
Benefit: When you cast a piercing spell against a target with spell resistance, it treats the spell resistance of the target as 5 lower than its actual SR. A piercing spell uses up a spell slot one level higher than the spell's actual level.
Combined with Spell penetration and greater spell penetration, you can effectively get their SR down by 9 (+4 to your attempt to punch through it, -5 to their SR for that spell).
Yes there is. its a spell called Spellbreaker. Its a 4th level spell with an immediate action casting time and in a 3pp called Eldritch Secrets
I think if you Death Knell a bunch of peons, your caster level will go up. There is also a rune stone that increases the your caster level.
SR is built into the game for a reason. Between spell penetration and greater spell penetration, bypassing SR should not be too much of a problem against most opponents.
Assay Resistance (and the various orb spells) were some of the worst aspects of 3.5 magic in my opinion.