First, don't worry about the bike - worry about you. If you need a different bicycle, you'll figure that out pretty fast while you're training to do your 200-mile ride.
What's the farthest you've ridden on that bike? 30 miles? Even 50 or 60 miles is only just getting to the edges of what riding 100 miles is like. Fatigue is cumulative - never mind just running out of intramuscular glycogen - all the microtrauma your muscles endure riding a mile is piled on top of the microtrauma endured in the previous miles.
Ever done 80 miles or close to it? Because you're going to need to work up to at least that just to be able to do 100 miles in one day without getting surprised by something like debilitating cramps at 94 miles.
Are you guaranteed to have something like that happen to you if you don't work up to the distance? No. But if that does happen to you on the first day of a two-day, 200-mile ride, what then? And issues like that are probably about ten times more likely on the second day.
To be able to do 200 miles in two days, an 80-mile ride in one day is going to have to be fairly easy.
When you first start riding distances like that, the final miles are hard. You won't just be tired, your leg muscles will feel like they're just about to turn themselves inside out, and they oftentimes will - grabbing at you with cramps that can literally make you fall over off your bike. Your arms and shoulders and neck can be awfully painful. Your fingers numb. Your back muscles spasming. You might get saddle sores, or worst-case, bad blisters on your feet.
Even if you work up to that distance slowly, there's simply nothing else that can prepare your muscles for the 6, 7, or 8 hours of continuous effort needed to ride 100 miles or so. Again - because fatigue is cumulative.
Coming back the next day and doing another 100 miles is going to be even harder. Your muscles simply will not recover in that short of a time.
So how do riders do it?
By working up to it. Over time.
So you need to get riding. Work your distance up over time - you have several months.
Go out this upcoming weekend. Plan on riding a bit further than your longest-ever ride - you should probably aim for at least 30 miles. Squeeze in 2-3 90-min or longer rides during the week. Allow for some off-days to recover. Add 5-10 miles per week to your long weekend ride.
After about a month of that - say you're doing 50-60 miles on Saturday - come back on Sunday with a 25- or 30-mile ride, really easy. Start adding miles to that, too.
Plot out your planned distances. I'd say you should probably aim for 70/70 back-to-back rides about a week or two before the ride.
Because it's that back-to-back that's going to be the toughest thing to handle.
And don't ride at all for three or four days prior to the 200-mile ride. By that point, all you can do is fatigue yourself.
And while you're working your distance up, there's a chance you'll discover all kinds of issues - your shoes don't work because when you get to 70 miles you start getting hot spots on your feet. (You'll know why they're called "hot spots" if you get 'em...) Your saddle might not work. Your hands might start going numb after four hours. Are those guaranteed to happen? Again, no. But they're all possibilities.
And note that I didn't mention your bike not being able to handle the distance at all. It won't have a problem at all as long as you do routine maintenance.
If that sounds hard, it is. But it's better than that 200-mile ride turning into a misadventure you'd rather forget.