You can definitely injure a gargantuan monster from within; it's actually the only way to get out of the Tarrasque once swallowed, so using those same rules (your character is blinded, restrained, has total cover, and is taking acid damage every round), it should be possible to adapt this to other monsters.
However, the fact that your character has total coverage may cause some issues, depending on how you intend to use magic. Coverage from attack is also coverage from help.
For more rules on swallow, I actually found a D&D Basic guide that contains some handy rules you could use. Large Frogs and Large Toads, while smaller than Gargantuan, have rules for Swallow. The Tarrasque, while larger than Gargantuan, has similar rules for Swallow. It is thus logical that a Gargantuan creature would have similar rules along those lines.