It’s not a sin to be intimate with your spouse if they insist on using contraception themselves. That, however, is not your situation. Instead of using contraception herself, your wife is demanding that you, rather than she, use contraception as a precondition for sexual relations. She’s asking you to commit a grave sin, effectively presenting you with a “me or God” choice. She probably hasn’t thought about it that way and is simply scared of the thought of having another child. Whether it’s just the financial concerns you mention, also the personal worries and stress of caring for another child, or any number of other reasons on top of those, the fact remains that she doesn’t consider herself ready for it again right now. And that’s fine. What’s not fine, however, is the method she wishes to employ toward that end.
First, I would recommend pointing out that, by insisting you do something which goes against both your conscience and Church teaching, she is effectively asking the man she loves to commit a mortal sin. I highly doubt that’s her intent. It probably hasn’t even crossed her mind. She just wants peace of mind and has the mistaken idea that she can get that from condom use.
Then ask her, instead of insisting that you commit a mortal sin, to attend NFP classes WITH YOU. Don’t make this something you’re trying to explain to her yourself. Make it an activity that you’re both doing together.
Also, don’t insist that practicing NFP is something she must do, no questions asked. Just invite her to learn about the program from people who have real world experience with it and see how it goes from there. Let her bring her objections to the NFP instructors rather than to you. Let THEM answer the questions. They’ve most likely not only heard them before, but are better at giving a clear answer. Most NFP classes are also taught by couples. If she’s only hearing about it from you, she’s likely to only get the “blah blah statistics blah blah effectiveness blah blah process” kind of answer that we males want to provide for every question. With a couple teaching, the woman can answer the question your wife is probably really asking (I’m scared; will you please listen while I tell you why?) rather than the question you may think she’s asking (What means may I employ to face and overcome my fear in the most efficient and effective manner?).
If she’s not willing to at least hear about the program from instructors, not necessarily to put it into practice in her life but to simply learn the facts about something that matters so much to her husband, she likely has much greater issues going on than concern over another child, and probably needs counseling more than NFP instruction.
My wife was hesitant to practice NFP until we signed up to learn it from the Couple to Couple League. My own suggestion wasn’t sufficient. She needed to hear testimony directly from someone else, someone who had been using it themselves for a time and could speak from experience on what it’s all about. If she had been the one approaching me about practicing NFP, I would probably have been hesitant as well. And that’s understandable. Testimony from personal experience can be a powerful thing. Just ask the apostles and the early Church
Whether you should use a condom between now and when you can arrange to take NFP classes… I don’t know. The question is probably best directed to your priest. Personally, I would choose abstaining over directly taking a sinful action. My wife used the pill before learning NFP, but that was her act rather than my act. She never insisted that I use a condom, so thankfully I never had to deal with that question.