Please forgive me if I've misunderstood, but what I see in your post are guesses and fear. What you need is to understand with far more certainty what is explained below as each side's BATNA. If you understand each side's BATNA, you eliminate the fear. There isn't a lot of room for fear in a winning negotiation.
Let's talk about BATNA. You can't negotiate well until you do the work (1) to find another few places to rent that work, including all expenses to up and move; and (2) to make sure you understand the landlord's likely losses if the landlord fails to renew with you. Your other rental option don't have to be perfect because perfect is for whinypants and you and your spouse will model for your kids that any transition is good. You also only need to be close on the landlord's potential losses, but the closer the better. You need both to know each of your BATNAs (best alternative to a negotiated agreement, see Getting Past No and Getting To Yes). The BATNA tells you what to offer, where you want to end up, how hard to push, when to say no, and when to say yes.
You only know some of the facts that you need to know to identify your BATNA. It reads as though you don't really know what other houses are out there and for how much. If your assumption is right that we're in a recession and people have less liquidity, you should find plenty of places for rent that fit your needs and for the right price. Given that the landlord is offering a month-to-month rent, the only additional rent you'll pay is the increase if you fail to reach an agreement to your liking. Your deployment only determines when you are available to move, not if. As soon as you are home, you can break the lease in I suspect 30 days.
The post also reads as though you like where you live and you don't want the inconvenience of moving elsewhere if you alienate your current landlord. Maybe you can save money elsewhere, and all-in with moving expenses, etc. you'll make your money back in fewer than six months. But maybe your family values certainty and continuity such that you don't care about the additional cost. That's a personal call that only you and your family can value. But the only real impediment to a better deal is fear and lack of information.
I'll tell you when you'll know that you have enough information. You'll have enough information when you answer the above questions, write a follow-up post, and realize in the writing that you already know what to do. Just be kind enough to pass it on by telling all of us what you did and why.
I hope that helps!
