@Robert Czek nice first post!
Did you ever think about what the best sport out there is, or maybe what is the best color? Well there is no best sport or color- there might be more popular ones - but everyone likes something different.
So while I can try to give some guidance to your question, there is no right or wrong answer to your question either - it's a matter of personal preference and what works best for your family. Remember some people will take serious risks in life, and others prefer a more conservative and balanced approach to avoid risk; and since you are different and your needs/situation is different than everyone else here, there is no one way or definitive answers (unless it's one of the scam artists trying to sell a get rich quick program - there is only one right way with them.) :)
Getting back to your question, which I am probably not qualified to answer since I am not a "smart" money person:
1) Your assumption was that you cannot get a second mortgage with your current salaries. Can I kindly challenge you to see if you spoke to MULTIPLE mortgage lenders and brokers to verify that? (I stress multiple, because some people see the world in black and white, and if you only go to them then they may have told you no, but...) I wouldn't be surprised if you can get a second loan with the income you stated. However, let's assume you cannot...
2) What about a Home Equity Line of Credit (HELOC) on your current home to pull out money to put more money down to buy the second house and rent out home #1? Is that an option? (Again, I would recommend talking to a local lender - ask around on BP for someone, it's amazing what some of these lenders can do.)
3) Let's say neither option 1 or 2 work, then it goes back to your comfort level. Do you like blue, pink or green, i.e. what is your risk tolerance for having a bigger mortgage? Do you feel comfortable covering that loan? If yes, and you want to move to bigger house then maybe that's the play. However, if you are you going to be up every night thinking "What if the roof blows off or what if the AC goes (that just cannot happen in Ft Myers!), or what if...?", well then maybe a bigger house isn't for you.
Again, it's a decision for you and your family to make and what your financial situation and comfort level is.
If it were me - not knowing anything else about you - I would try my best buy home #2 and see if renting home #1 makes financial sense and get into the real estate investing game. But again, do you have the tolerance for that? And you may think very differently than me.
I guess there actually is a best and worst color when it comes to net profit: staying in the black is the best and dipping to the red is the worst.
Best of luck!