mist4life,
Our previous coach, was a '99 Fleetwood Bounder 34V. It was also on the "F-53" chassis, not the "550". It had the 275HP V-10 and whatever trans that came with that era Ford. It took quite a bit of responses but, Richard nailed the correct description of what you're hearing and seeing when you energize the ignition AND put your foot on the brake pedal. It's a power brake pump, not a trailer brake assist. If I'm not mistaken, it's the ONLY power assist that is used for your brakes but, It's been a while since I've owned it so, I could be off here.
And, as has been stated, with only 6K miles on it, your brakes SHOULD be in outstanding shape. That is, the SHOES or PADS, whichever it has. That too I've forgotten. We had our Bounder for 7 years and it never, EVER let us down. We towed several different Jeeps from San Diego to Moab UT for years with it. We really liked it.
Anyway, it's advisory that ALL the brake fluid be exchanged in those chassis's every two years. Not many follow those type recommendation but, again, it's advisable. I did all the maintenance, alterations, modifications and more on that coach. That was a great engine. If I recall, the coach came with Power Gear levelers. They too, never let us down. I did have to rebuild one or two of them due to some slight leaks. That was also, no big deal.
I had to rebuild the parking brake, called a "ZF" parking brake. It's the kind that's attached to the rear of the transmission and uses a brake drum. But, because those engines are fairly powerful, you can over power that parking brake and forget it's still on. That's not good 'cause it burns up the shoes and get's the auto tranny fluid that lubes the bearings inside that ZF really hot which, is not good on the seals. So, once you get this new-to-you coach up and running and are using it, DO NOT forget that parking brake is ON.
As for your "shaking" at 45 mph, well, there's potentially a few things that could cause that. I'm no expert but, the suggestion your brakes might be doing it is, well, I'm not sold on that one. The reason, why would brakes do it at ONLY 45 mph and not before or after? If, as you say, it's got new tires on it, I'd aim at two potential causes. One, one or more of those tires may not be properly balanced. And two, one or more of them could, I say COULD be potentially out-of-round. And if that's the case, you could attempt to balance it all day long but, an egg is an egg, you can't balance an egg. There's only two ways to check for out of roundness. One, on a balance machine while it's turning and observe its rotation. You can see it with your bare eyeball if it's bad enough to cause your issue at 45 mph.
The second way is, well, it's done on the coach. If on the front, you'll need a seasoned vet in tire balancing and old school tire spinning machine. That will show you if it's out of round. If it's on the rear, that's a bit tougher to do. One has to be real careful spinning up one set of rear duals while the coach is in the air. Done it a few times, ain't fun.
Anyway, those are a couple of suggestions for your shaking.
As for your battery situation. Well, I don't know what arrangement you have for coach batteries but, I went with (2) Trojan 6V units for house batteries and, they worked great for about 5 years 'till I had to replace them. And, again if I'm not mistaken, you SHOULD have what's called the "BCC" or, Battery Control Center, under the front part of the tilt-up hood, about just in front of the driver. That's if things haven't changed between the '99 and '00 years. That BCC is full of fuses, relays and more that control just when each set of batteries gets its charge, from the engine, depending on the state of charge, for each set.
For instance, if you've been camping for a while and your batts are low in charge, then you start up your engine. Well, that BCC will relay the charge from the alternator to ONLY THE CHASSIS BATTERY until it reaches over 12.8V. At that time, it will then send some of the alternators charge to the house batteries. The chassis battery is more important than the house batteries so, that's why it does that.
Anyway, do not give up too easily on this. It's a great coach and will do fine for you for years to come. Feel free to PM me for any questions and, I'd be glad to help, if I can.
Scott