Not sure why you're being considered infertile even though you've been trying for less than a year. Here in the UK, a GP won't bother with you until you've been trying for over a year - you certainly still have hope. I've read a lot of times, 'infertility' is caused by bad timing - me and my girlfriend used this ovulation tool
alongside ovulation predictor kit sticks you can get super cheap off Amazon.
Other than that, I feel like you two just need a serious conversation to discuss both sides here, how does she feel about being sexless, what's causing it, etc. Of course everyone's different but I'm really not convinced jealousy tactics will help at all in this situation.
Hope everything goes well whatever you decide to do