Recently been playing an Battle Master in a campaign and was rewarded with some spellwrought tattoos, and one of them was for Silence. A pretty great spell for fighter I figured, since my Athletics is pretty damn good so I can grapple enemy casters into a zone of silence, and spell wrought tattoos can be used by anyone not just spellcasters. And while the opportunity hasn't come up yet, I have been wondering if it's possible for characters such as Fighters who have access to these magic tattoos, or more likely just Eldritch Knights who have access to magic in general, to use their bonus actions or Action Surge to cast spells between their attacks, if they have Extra attack.

The hypothetical plan I had in mind goes like this. My level 5 polearm master fighter vs an enemy caster. I go in, hit them with my spear. Bonus action attack with the other end of my spear. Then, I use an action surge to activate the tattoo and create a zone of silence around me that also covers the mage. Then I finish my extra attack with a Grapple, winning the grapple vs the mage's athletics/acrobatics. Enemy has taken damage (with additional damage from my maneuvers, such as Trip and Menacing), is grappled, and is now in a zone of silence.

The reason I want to know if I can cast the spell in between attacks instead of just casting before or after is for tactical reasons. Specifically, if the mage has counterspell, I want to try to make them waste their reaction with a Shield Spell instead with my spear and bonus action attack, since if I do hit I can do extra damage with maneuvers. If they do take the bait and use Shield, then I can safely use Silence without fear of them counter spelling since they already used their reaction. And even if their AC is still high, that wouldn't matter for a grappling attack, which is a contested check not an attack roll vs their AC.

Another reason why I want to know if I can cast between attacks instead of simply doing all of my attacks first, then action-surging to spellcast, is that if I do make a regular attack, bonus action polearm master attack, and then a grapple attack, I'm not sure if I can cast my spellwrought tattoo, since presumably I still need a free hand to do Silence's Somatic components, which is hard to do if one hand is holding a regular spear and the other hand is grappling the enemy.

Thus, in order to bait out any possible counter spell I need to make some attacks first, which ideally one might miss because the enemy mage uses shield. Then I can have a free hand to cast Silence off the Spell Wrought Tattoo, and then finish my turn grappling the enemy.