Given your start date, you'll be hitting the southern portion of the MRT in the midst of heat and humidity season. We're talking 90+F and 85 or better relative humidity. The problem with rain gear at those numbers is you have a choice - get wet from the rain or from being inside a sauna (the rainsuit will not allow sweat to evaporate/escape). So, being wet is just one of the issues you'll face - overheating and heat injuries are a bigger threat to completion, imho.

As a Lousyannian for decades and having spent 3 years in the St. Louis area, trust me, once you reach there, rain is probably NOT your primary concern. The chance of the temps dropping below 70F south of there is minimal and then, only at night for maybe an hour if that. North of there, well...ask someone else.

As for flats, I don't see how fixing a flat on a BD Windsor Tourist would be any different than doing so on any other diamond frame bike. It's got a 700c wheel and a derailleur. Big deal. Just do it.

I recommend bringing a GOOD pump - something along the lines of a Topeak Road Morph. Small. Ultra-reliable. And it has a built-in gauge. Check eBay for decent deals. I got mine for $15 though it regularly sells for much more. Practice patching at home before you leave. "Wasting" a few patches in a dry setting to learn will benefit you later while on the road in the rain for a really low cost. Seriously. Learn how to patch correctly and you'll save yourself time and frustration later. (Ex., I learned to NOT spread he glue with my index finger - but instead with my little finger... Why? I don't NEED to have "traction" with my little finger pad but often do with my index finger(s)!)

And practicing taking off and putting on tires at home will give you confidence while possibly identifying that your technique for doing so causes pinch flats or requires 3 brake levers not just one or two.

Last piece of advice --- water. Take 3 750ml water bottles and/or a hydration pack. Even though towns may be "easily reachable" mileage wise, the heat in late May/earlyJune in So. Missouri, Ark and Loiusiana can be brutal!. If you feel lightheaded, find shade and drink drink drink. You've heard the desert is a "dry heat"? Well, the Deep South is a wet and steamy one. Sweat doesn't evaporate in a humid environment. So, no cooling. Trust me. A little more weight due to carrying water may save your day if not something more valuable. I carry 3 bottles and/or a 3L hydration pack every ride in the summer in the New Orleans/Baton Rouge area - and that doesn't count the Gatorade I may also be carrying. (Of course, I've been called conservative --- but I've also suffered heat injuries by trying to tough things out.)

Oh - almost forgot --- sunscreen. At least 40SPF. Slather it on no later than 9am daily and every 1.5-2 hours throughout the day. You WILL burn otherwise. Badly. Even on overcast days.
(Some people wear loose long sleeved shirts and/or long pants instead.... your choice.)

Obviously - YMMV.