Unfortunately, you cannot care more than the bio parent.

Also, I would have a hard time respecting my partner if they were being such a lazy parent with their kid.  Do they take him to the DR for regular checkups?  hasn't the DR raised some alarms over the weight etc?

When he's with us, we have to make separate meals for him. 

I am not a restaurant chef. Not we -- DH. I'd be cooking normal meals. If SS won't eat them, fixing a meal for him is his father's responsiblity.

Or it's SS's responsiblity. SS can fix his separate meal after we have eaten and the kitchen is clean. After SS is done eating, he can clean up whatever he used.

(And I will not be picking up the ingredients for SS's separate meal when I do my grocery shopping. That's up to his dad.)

I went through the same thing with 3 SD's and DH. SDs wanted to eat junk and wondered why they felt awful all the time. DH didn't wan't to spend his 2 days with his kids every other week fighting over food, so he gave in. 

Now they are all having health issues. They have a cousin who is obese for the same reasons at 8 years old. Maybe one day they will figure out that I was trying to help them by offering healthy foods. Not my problem anymore because I got tired of caring more than the bioparents. 

I would let them handle it.

When he's with us, we have to make separate meals for him

If there is not a medical limit to his diet, you don't have to do crap for him food wise that is any different than what everyone else in the home eats.

As for screen time, turn it all off. End of problem. We banned video games from our home when SS-31 was in 6th grade. Gone. Zero.  End of problem.

As for mental health days granted by mommy, call the truancy office and make anonymous reports on that crap.

My ILs did the "we let out kids make their own decisions" and "we raise our kids to be independent when they turn 18" bullshit and failed in 75% of their children.  The only one who is a high performance fully viable adult is my bride, the oldest of 4. Bil1 barely graduated from HS and is a farm laborer approaching his mid 40s.  BIL2 barely graduated HS and is a farm laborer in his early 40s. SIL is late 30s and a crook who will steal you blind in a split second.  She "went to college" has no degree and over $100K in school loans.

Pretty much par for the course for the "let them make their own decisions" and "raise them to be independent at 18" crowd. No standards, no boundaries, no performance requirements.  They end up with screen brain and arthritic controller thumbs by the time they turn 18.  They have to take their parents on job interviews with them. While they are in HS they have to have their parents go scream at or beat up their teachers.

Get ready, this "make his own decisions" spawn will be a blight on your life and home forever because the odds of him every being self supporting are slim and none.

NO..just say NO.  NO is actually a sentence.  DH can cook what ever he wants for you and SS.  He can do the grocery shopping and the cleaning up. 
With both bio parents buying his unhealthness. Nothing is going to change.  He most likely has no friends, unless online, so what do you expect.  
The Bigger question for you, how do you respect a person like your SO.  he just about the worst father there is.  Instead of getting his DS into sports,  team mates, friends, something in common. He lets him vegetate in his room.  That's a big character floor.  You want kids with this person? 

Oh, hell no. To all of it. 

Read my blogs about OSD(now20).

Refused all fruit and veg and ate only about 7 or so different foods, mostly white; would starve rather than eat a "rejected" food (this is an eating disorder ARFID, take a look at the description of it, it's more than just picky eating...I did the at-home food "test" with her when she was younger and everything matched with her, her sister was just picky). DH was cooking separate meals for the SDs when I entered the picture, however I refused to do so and helped create ONE meal we could all pick and choose from (think spaghetti and meat balls broken down into separate bowls so components could be put together as the person wanted, or tacos, or pizzas). Never told no until she was 13 and she freaked out (also PAS was involved). She had a free pass online, DH maybe checked her internet use once or twice, totally ignoring work-arounds teen used - i.e. he never checked her actual phone; she'd be up until 2-3 am and exhausted, but still can't take baby's phone can we? I tried starting chores charts and allowances that worked until DH agreed with OSDthen11 that it was "too hard."

I can go on.

How it ended?

I am totally disengaged from both SDs. You cannot care more than the parent/s. You cannot DO more than the parent wants or allows you to do. I had many, many discussions with my DH and he would simply NEVER ever allow me to "parent" - he wanted me only to support him and do it his way (passive parenting) and provide help when he wanted, and entertain/give them nice things. I could not do that. So I handed everything off to him incrementally.

OSD broke from DH in a horrible PAS situation when she was 13/14 and he's not spoken with her since she was 17. She's an entitled, manipulative, mean person. YSD18 still maintains contact with DH, but she's also entitled and rarely thinks of others and is just an odd person (I think slight autism but what do I know aside from being the only one that's done extensive reading on girls/autism?). It's not ended well for him and he has a lot of unresolved issues that do effect our marriage.

  Don't make another meal for the kid.  Kids need to learn that families all eat together, and eat what is made.  If they really cannot eat it, they can go to the kitchen and make themselves a sandwich with what is in the kitchen, and then tidy up after.  It's not fair to treat you like a short order cook! 
  Your partner needs to step up and take the screen away. All day long is ridiculous. There will be tears and tantrums, but as of now it's not the kids fault, and parents need to step up. 
 
   

My partners 2 kids are constantly online and he does nothing to moderate it. They aren't too fussy but will sulk if they don't get what they want for dinner then sometimes pretend they're not hungry so that they can ask for toast or something a few hours later (and their dad often agrees, much to my dismay) 

tip on the gaming- I have paused the internet connection at my house which boots them off mid game. I pause it for 2 hours at a time and overnight it is shut off from 9pm (offline from 9 is the rule) 

At first they were annoyed and bored and couldn't figure out what to do - when my suggestions were clean your room, go for a walk, go to the skate park they started to find other things they couldnoccupy themselves with. 
 

while this internet rule also affects my partner and I - it gives us the opportunity to connect and spend time together without distraction. 
 

i think of it like this: it's my house as much as my partner, and rules apply to us all. No exceptions.