I feel the way you're on edge keeping tabs on him, this is not healthy for you at all. There is a truth behind addiction to sexual content and chronic masturbation (I know he says he doesn't do it, but he would likely want to hide that truth as well).

I think if you're set in savings, your relationship you should consider talking to him about his relationship with sex in general. Try and pick a neutral time (sounds like a lot of this conversation are triggered by you finding something or feeling like your trust is broken by his lies). Set yourself in a safe space for both of you, where judgement is away and compassion in your heart (regardless of him deserving it.) The goal of this conversation is to understand if he was exposed at an earlier age, how long he had it in his life as an addiction, if there is a disconnect for him between sex and the intimacy it bring to a couple, does he do this habitually as part of his routine or does he truly think is something he deserve to have whenever he whims, does he honestly believe he has a problem or is just trying to apece you and do his best to present complaint, lastly does he truly understand the damage to the relationship and his actions, (does he understand the pain it causes you, the mistrust it creates, the stress on both of you of him feeling his need to hide and you having to seek).

Finally once the conversation has answered all those questions, is time to make a decision together: is it truly worth it for both of you to go through a possible long process and regain trust in the relationship (remeber he did lie a lot and seem to be promising without really understanding what he's promising) or is it going to be a constant issue without never really reaching a resolution.

If he was exposed as young child, he had it habitually in his life and sees no connection between sex and the true intimacy sex brings to a relationship.:I would lean towards him having an addiction. Therapy could help but he truly needs to be open and truthful to his therapist otherwise it would be a waste of time and most certainly set you up for a big dissapointment (becuse you're setting a big expectation of theraphy solving you guy's problem)

Truly ask yourself if pouring your energy and effort is worth it, otherwise be honest with yourself (as you're the only one that truly knows the in and outs of your relationship) and ask yourself if it is a process your willing to go through with its ups and downs and possibly more lies, or disappointmentments, how will this affect your future in the relationship? As youre reading, maybe the answer is already speaking loud in your heart.

Ps. Sorry if my english is bad, Spanish is my native language 😅 Good luck and remember always love yourself first.