The colour palette is really snazzy. I dig games that use a limited palette to their full potential like Downwell or
ZeroRanger. I would caution that the player sprite's brightness is very similar to the background elements, which could cause visibility issues for players with vision issues like colourblindness (consider a high contrast player sprite option or colour palette options). Most 8 bit shmups deliberately make the player ship either bright, or highlighted with white elements, or will use a colour that is reserved entirely for the player and the player's own shots to maximize visibility.
Agreed with the sidebar art. Jet Buster is a lot of fun to play but I would not say its art is a highlight of the game. In your case I think the art style very slick, and the "furry" elements shouldn't immediately deter anyone, but generally sidebars that feature anime characters and whatnot are not as appealing as more minimalist, lest distracting ones. At least don't make the sidebar art default to them and you should be fine.
Don't feel bad about making an RPG shmup hybrid. It is possible to make them work; Guardic Gaiden/The Guardian Legend does a decent job of it, Tyrian 2000 does too. However, both those games also feature
pure shmup/arcade modes that drop the RPG elements (in Guardic Gaiden's case as a password you get for beating the game normally) so consider that as an idea for satisfying both a more casual audience as well as appealing for more hardcore shmuppers. It depends on who you're wanting to play the game.
Not sure how much screen shake there is, if any, but screen shake is horrible and has shown up in a number of recent games I can think of without the option to disable it. This is quite unfortunate, so consider having the option to disable screen shaking if it's a thing (Downwell would be infinitely more fun without screen shake, Monolith certainly is better with it turned off since most of its weapon cause shake every time you fire them, ick, Score Rush Extended is a rare exception that's fine because
you're invulnerable during screen shake so the shaking is actually more of an indicator rather than a hindrance).
I'm just worried it will make it harder for the player to see their own bullets if there are a lot of enemy bullets on the screen.
The thing that always, always needs to be visible at all times are enemy bullets. If you're firing a constant stream of shots, the player shots will be visible enough, but the priority on visibility should always be on enemy bullets. It's far less of an issue if player shots are obscured slightly by enemy shots because you can use your position onscreen to know where you're firing, but if enemy shots get obscured then you're getting hit by things you can't see, and that is
Extremely Bad in a shmup. For an example of this, look at the windows-era Touhou games like Imperishable Night; the players shots appear to be partially transparent, and are much less bright compared to enemy shots. The focus is entirely on maximizing enemy attack visibility, but because the player's shots move so rapidly and fire so quickly, they contrast well and you can still see them just fine distinct from other elements.
Generally speaking, the order you want to layer things is:
Enemy Bullets
Player Bullets
Enemy Sprites
Player Sprites
This way if the player has a small hitbox relative to their sprite (something like Battle Traverse is an exaggerated example), the enemy bullets won't be covered up by the sprite and hidden while they're still far enough from the hitbox to dodge. Judgement Silversword is an example of a game that is fantastic, but has the unfortunate design flaw of having the player sprite be ABOVE enemy bullets. The hitbox is just small enough that it's possible for a bullet to go under your sprite but still be in a position you can dodge, resulting in some visibility issues during dense, slow attacks (Mirror Shield boss in particular).