Usually when developers do a 'purchase the game for (tier) and make (in-game bonus)', things like obscentity or advertisements in-game don't get included, but there are ways around that. Like having a character involved in the mission named 'Raymonddog' or 'Raymond Dog', or including a link in the credits outside of the game. But I could be wrong about that.
The best way you could promote yourself in a special mission, I think, would be to request a special Arena mission where you're the announcer, Mr. Twitch Raymonddog, or Raymond "Twitch" Dog, maybe? That would put you prominently in the role of the mission without making it an obvious advertisement.
By being at the arena, it would be a graded mission and therefore one people have to try over and over to get double S-Ranks in and repeatable, giving you more of a presence.
For a hypothetical arena challenge? You could probably design a cool one, several bosses or vast numbers of elite enemies that have HP/stats adjusted to match the player level, each enemy given the name of your Twitch channel moderators, subscribers or users. In place of the usual 'Slime the Mad', for example.
If you want to design a whole area for your mission instead of just having an arena challenge, you could request a mission themed after your favorite genre of games.
You like zombie games? A haunted house or crypt filled with the shambling undead, monsters who look and fight like people. Give them all large bones and small bones to fight with and have each one auto-cast a specific melee skill every few seconds while homing in on the players. So one shambles over Whirlslashing every few seconds, one Tosses their weapon every few seconds, one goes Berserk and gets a damage bonus, one casts Spirit Slash, one Piercing Dashes, etc. Each zombie in the area could have a name of you or prominent people from your twitch chat.
How about side-scrolling beat-'em-ups? You could request a special area that's not very tall at all, but extremely long and only scrolls to the right (ex: Golden Axe, Legend of the Mystical Ninja, Castle Crashers). Waves of enemies and bosses from the game and other obstacles stand in your way. How hard is Gigaslime when you can't run up, down or back? Or being tag-teamed by a powered up Marino and Vilya at the same time? If you make it a dream area or an area of illusions, you could take any liberties you wanted with the bosses present since it wouldn't conflict with the storyline.
FPS games? Put in a special area where your weapons are as heavy as boulders and your magic doesn't work. You have infinite arrows and have to Rambo your way through it, sniping down everything in your path.
@Teddy absolutely hates anything bow-related, though.
Tower Defense? You have a set path, or several paths, enemies can travel along at full pace towards the goal. For every enemy you let slip by the end takes damage. You can attack enemies to distract them into attacking you instead of racing for the goal. Between waves you can stand on squares along the paths to 'grow' specific plants up like you would normally revive partners. The first growth, the basic plant. Then you could stay on it even longer to grow it into second charge, third and even longer for the final gold charge. That might be a unique idea for a mission. Again, enemy names could be people from your channel.
Roguelikes? You could ask for a special dungeon like Arcade Mode / Binding of Isaac, but in Story Mode. A randomized # floor dungeon filled with powered up versions of enemies from all across the game, not kept apart from where they are in Story Mode. A room with 2 crystals, an elite boar and a couple of ghosts and an Autumn Hydra head? Why not.