I've got a couple of APC ES 550 UPS for my desktop and Google fiber modem, mainly to allow a graceful shutdown in the even of a power outage. One of them is telling me that I need to replace the battery, which is a 12V 12Ah battery.

At this point, I see that the Lithium FeSO4 battery in this size is now about cost equivalent to the sealed lead acid battery. But they support many more charge/discharge cycles and lithium tolerates deep cycling much better than SLA. The lithium batteries all come with a battery management system built into them for safety.

Any reason I can't swap out the SLA battery for a Li battery? Assuming the BMS is doing its job, then charging and discharging should still occur just fine, right? Right?

Any experience here?

For reference, I'm talking about replacing this:

https://www.amazon.com/ML12-12-Battery-Terminal-Mighty-Product/dp/B00K53FG5Q

with this (or similar):

https://www.amazon.com/DC-HOUSE-12V-12Ah-Lightweight/dp/B0CNCQGX3D  

I did something like that with my APC UPS about a year ago. I purchased a different brand of LIFPO battery from Amazon.  No issues while line power was available, but at line power failure the UPS made this high pitched warning tone, but it still powered our computers and peripherals for about 5 minutes before I shut everything down. I had to press the power on/off button to off to stop the annoying screaming tone. YMMV

They always make some kind of noise when they lose power. That's expected. I'm just hoping for longer duration and better discharge performance, while not burning down the house.

LiFePO4 BMS typically need a higher voltage to do their job than a UPS designed for SLA will put out.

Many UPSes maintain the SLA using a float charge, typically 13.5-13.6V. LiFePO4 batteries are not
float charged, and typically need a charge input of 14.6V for the BMS to do its job and manage a full
charge on the battery. Without the higher voltage the BMS will never be able to do a complete charge
to 100%.

With the undervolt you can't really predict what the BMS will do, it may try to charge the battery at the
lower voltage or it may simple not charge it at all. Some LiFePO4 batteries will do OK in SLA applications
like small engine batteries because the voltage regulator will reach the voltage needed by the BMS, or at
least a voltage much higher than the float charge voltage, resulting in a higher state of charge, but
in a UPS application the results will be unpredictable.

That's the charging side. The discharge side is also a concern depending on the maximum current rating
of the LiFePO4 battery that's replacing the SLA. SLAs can briefly handle discharge currents well in
excess of their capacity -- typically about 3X capacity for 5-10 minutes (e.g. a 7AH SLA can supply 21 amps
for a few minutes) which is one of the reason UPS manufacturers pick them. The BMS system of whatever
LiFePO4 battery you pick will need to be able to handle high currents, and many off the shelf replacements
for SLAs aren't up to the task and will drop out, or worse actually melt inside the battery casing.

Your best bet is to find a UPS designed to use LiFePO4.

ETA: Using the batteries you cited, the might max SLA will put out about 36A -- more than 2X the 15A BMS
rating of the DC House battery you linked. The UPS load will need to be kept below about 150 watts to not
go over-limit on the LiFePO4, and of course that doesn't address any potential charging issues.