So sorry to hear you're going through this! Step-parents are always taken for granted and harboring the perfect blended family is not an easy job but it can be worth while in the long run. My brothers made it difficult for my step mum when they were younger and adore her now (so there is hope!).
For me, it sounds like the issue lies more with your partner rather then your step son. Your partner needs to support you and visa versa, maybe spend more time with him and open communication, be honest and non confrontational? And if he feels your relationship with your sons Dad is too close then back off a little, it's great to have an amicable relationship with your ex partner/ biological father of your son but it's not necessary to be buddies at all, especially if it upsets your current partner which later on affects his mood/behavior in the house consequentially affecting your son. You could speak with your ex partner as you would do with your child's teacher, keep it amicable but focus entirely on the child.
Would it help if you pushed yourself into the frame mind of: this is not my step son, this is MY son. Weather he moans about me, acts ungrateful or does things I don't like. I'm here as a care giver and I'm not going anywhere.
When you see them as a 'step child' that you're thinking about 'calling it a day on' you can almost fall into the wrong mind set of not wanting him around like it's an option.
It really sounds like you're trying hard and are the back bone of this blended family but if you could just re-train your brain to view your step son as your son, permanently, forever. It might help you overcome any difficult behavior.
Whenever your step son complains to your Dad about you, take a second to listen to what the complaint is and see if you can work on it and if it's just a rant with no basis, then get your partner to remind your step son how much you love him and talk about him all the time, love him to pieces and never want to upset or annoy him?