Which bank are you going through? I just closed on my first home via FHA, and they never even asked me about rental payments and I've been renting for years. My credit file has been around for awhile, but was completely filled with baddies until lately. I even had a judgement and reposession in within 3 years. I applied with my credit union's mortgage center as soon as I hit 620 and was approved. Closed on 7/12. If I were you I'd search around...
What CC do you have? The FICO scoring model wants to you have at least one CC from a bank (WF, BofA, etc.) or national credit card company (AmEx, Capital One, Discover, etc.). If your CC doesn't meet this requirement, you would likely benefit from getting one that does. It also wants to see periodic usage. A good strategy is to use one card for a small purchase every month and pay in full when the statement arrives (before any interest accrues, but after it reports). Best to keep the balance below 10% of the CL, or at least below 30% (never allow a balance of 50% or more to be reported). The car payment (installment) loan is also helping your scores. You would learn a lot from reading The Road to 850 (second edition 2011) by Al Bingham. He's the most knowledgable source I've ever found on improving your FICO scores.
Definitely start using checks ASAP. I used to think that using cash was the only way to go for all kinds of things, but have since learned that having a cancelled check is a much safer way of doing business. I went FHA and I had to provide all kinds of proof for the rental. Fortunately, I had deposited the cash directly into the landlord's account, so kept those receipts as well as the receipts showing my withdrawals. I had to submit more bank statements than I would have had I simply used checks which caused me to have to write more LOEs for some deposits. It was a pain.
The part of your post that jumped out at me was "I'm not sure we can manage a two year wait". The one thing I learned through this process is PATIENCE. Regardless of what I felt I NEEDED, it simply wasn't going to happen if I didn't get my financial ducks in a row. It's best to focus on the things you can change (using checks, building credit) than the things you cannot (denials, which are also negatively impacting your credit)