More power to you, raising 5 kids on a very middle-of-the-road income!
You need to run the numbers carefully, but the rent-vs-buy calculation is pretty marginal given that a 5% down mortgage puts you in PMI territory, which can add a solid point to your mortgage rates.
From a prudence point of view, I would not recommend buying until you had the down payment, plus closing costs, plus an additional $5,000 to $7,000 contingency saved up. That's moving costs, repairs and refurbishments to the new home (when the roof leaks it's now your responsibility -- no landlord), etc. Talk to some bankers about closing costs -- $10,000 for that line item may be a bit too high. But I still think you'll want $24,000+ in your house fund before you pull the trigger. I wouldn't set any loftier goals than that.
You could Roth-convert the traditional IRA, but then you need to wait 5 years before you can pull the principal amount out tax and penalty free. And given that your retirement accounts are not very large, I wouldn't recommend raiding them -- you want something put away to grow for old age, plus they are additional emergency funds, given that a $2,000 cash emergency fund for a family of seven is not at all large.
With respect to your budget, you have a bunch of things under "discretionary" which aren't very discretionary. You need to get to work one way or another. You need to buy groceries. There may be some variability or some luxury in those categories, but I'm not seeing thousand-dollar line items for "designer handbags" and "European vacations," so I don't see that much discretionary spending there.
A budget of $36,000 (including rent) for a family of 3 is a frugal but comfortable one if spent wisely. Scaling up to a family of 7 and a budget of $48,000 is a great accomplishment. I don't see much to criticize on the spending side, apart from the 240 mile per week commute. You're using the right car for the job (I'd expect your expenses to average $0.25 per mile rather than the U.S. average of around $0.50), but that's still basically $60 per week. If you could reduce that (change houses again; change job), you could likely save $1,000 or $2,000 per year on commuting costs (gas, clutch replacements), which would make a difference.
To make your dreams of homeownership come true reasonably quickly you really need to think about the income side. Can you upgrade your job? How old is your youngest -- can your wife starting accepting more projects once all the kids are in school? You've said very little about that in your post, but that's really the key.