Haas battery and Real Time Clock replacement
Gonna put everything I've learned in this post so people in the future don't have to sift through everything. I believe this applies to the Haas controls from the beginning of the vector-drive and brushless motor era: around 1997 to 2006?
Batteries: one or two?
The difference here seems to be the amount of memory your board has. I have not verified this. My personal sample size is two machines, plus what I can make out in pictures on eBay processor board listings. My 16 MB board from late 2004 has 32 RAM chips soldered to it. It has two backup batteries. I can confirm that by the original design, the two batteries are not connected together so if power is not supplied to BOTH pads, some of the memory will be lost.
A friend's Mini Mill from first quarter 2005 is only 1 MB in the form of two chips on the board. It had pads for up to 16 chips and I believe this is the same board up to 8 MB. That board only has provision for a single battery.
Remote Battery
Haas makes a very nice and easy to purchase remote battery that can be installed outside the board stack. Once installed, it can be serviced without future disassembly. It plugs to the EXT BAT pins above the original batteries. Installation involves plugging in the external pack, clipping the zip ties on the original battery(ies) and then clipping the soldered battery leads from the board.
Haas has a known issue against this pack. On these pre 15" LCD machines: a Low Battery Alarm will remain after installation. The only way this is possible is the board is checking to see if power is coming from the EXT BAT pins vs the battery pads. Haas may tell you the pack will cause this error and they can't fix it, though it will function fine.
To fix the error: clip the wires off the end of the battery lead and hard wire to the board. If you have a single battery board, it's two solder joints.
If you have a double-battery board, you need to add a jumper between the + Positive pads to provide 3V to both banks of memory.
Obviously, all of this should be done with 3V applied at the EXT BAT pins or all the control settings and parameters will be lost.
A Cheaper Battery Alternative
$95 too steep for a couple of external batteries? Head on over to Digikey and order a Keystone Electronics 1120 dual battery socket. At the time of this writing, it will set you back less than $5. Its a plastic case and and does not need to be mounted to a circuit board. It has solder tabs and can be connected any way necessary.
I use 3M double stick auto body adhesive tape to mount this case. I mount them either above the DC power supply at the top of the back wall of the cabinet, or on the left interior side wall. For security, you may opt to wrap a zip tie around the installed batteries. I did this to a friend's machine because it's about to travel 800 miles to his new home. I have now done four machines this way.
For our application, we want two cells, wired parallel. That way we can change one battery at a time and not risk losing data. Solder leads from Positive to Positive and Negative to Negative (shown below). Finally, solder 12-18" of wire to reach the pads on the processor board.
The batteries are available from multiple brands, though Digikey sells the BR2/3A cells that likely came in your machine for $5.10 each. Except for wire and solder, this entire option should cost around $20 shipped.
My Clock has a Bogus Time and Date
This is how I found this thread: I'll call it the Back to the Future bug. My Mocon had an issue and I had to dump all the alarms to floppy to review them. I discovered that all the errors for quite some time, had been time-stamped 06-26-2066 at 26:66:00. Replacing the battery(ies) did not fix this. If I reset the clock, it would revert back to 06-26-2066 after restart.

As discussed up-thread, the problem is right next to the batteries. There is a real time clock and calendar chip, in a socket and held by a zip tie. It's the Dallas Watchdog DS1286 real time clock. The chip has its own battery inside the shell. The battery had an expected life of ten years. After that, you were supposed to replace this chip. Unfortunately the chip has not been supported in this form since about 2008. Many computers, industrial machines and pieces of test equipment used this chip and method for time. Lots of related videos on YouTube from these other computers and equipment.
If you poke around the web, there are procedures for using a dremel to grind through the side of the chip's case, gain access to the battery leads, where you then clip one of the leads and solder external jumpers to connect an external battery. This is possible. I didn't do that, though you are more than welcome to look elsewhere for the details.
I found a seller on eBay who has a small side gig making replacements for the DS1286 chip. He uses the surface mount version of the same chip. He has breakout boards made that adapt the pins to fit the form of the 1286 and allow it to plug directly into our processor board. It features a very large, external battery. The battery is not removable, though I suppose in 10-15 years, you could desolder and replace it. You could optionally clip the battery off the board and wire it over to the same 3V pads as the rest of the machine's memory. I opted to install it as supplied since I can now get to the main board and reassemble it in less than an hour.

To swap in this replacement chip: clip the zip tie around the old DS1286 chip. Gently pull on the chip and it will come out of the original DIP socket. Carefully align the pins of the new board, paying attention to the position of the #1 Pin (should be at the upper left corner).
He sells this kit for $50. I have no affiliation, other than it fit nicely and avoided the kludge of dremel and solder blobs on the side of the old chip. You can find it on eBay by searching for: Tektronix ds1286
I have forwarded the relevant information to him and he may soon be selling these under the Haas keywords as well. I could have used all of this information two months ago. I hope recording it here will help others.