I decided to buy a few "smart" or "intelligent" battery chargers and try them in lieu of a wall-wart or brick. Here's what I found:
First, bear in mind that my equipment is drawing as much as 4A peak, so many of the typical one or two amp "trickle" chargers won't work.
Second, most of the smart battery chargers need to sense some voltage, usually at least 2V before they will charge, so having a battery or maybe a capacitor on the circuit is essential to make them work.
Third, the LiION battery I have already has a complete "intelligence" circuit with protection and charging devices built-in. It is designed to be charged by a wall-wart and does not need a proper battery charger.
Chiefly, I tried a NOCO Genius 5. This is a 5A "smart" charger. I also tried a couple of knock-off brands that were half the price. One looks to be a Russian design made in China, and the other is all-China. Both these latter chargers have rather loud fans.
The Russian job is labeled "Foxsur." It's too loud to power my pile of Wifi and Pi stuff, but it has a feature to "pulse" the charge (voltage) to desulphate and destratify batteries. I used it on an 12-year old AGM battery I have in my pickup truck and it brought it from a OCV of 11.8V to and OCV of 12.4V, tested at least 24 hours since it was on the charger. My regular NAPA battery charging cart could not do that.
The NOCO similarly has a "repair" function to desulphate/destratify. I used it on a 6-year old AGM battery that had an OCV of 12.1V and brought it up to 12.2V -- it did not seem to make much difference, maybe because the battery was not that bad to begin with.
I use the ~$300 Odyssey AGM batteries in my vehicles because they give better performance and last two or three times longer than the flooded lead acid batteries. They cost a similar factor more, but since vehicle battery failures are potentially grievous, I don't just go by $/year. For my amateur station equipment, I am likely to use a spare automotive AGM battery because I don't get much value out of a battery just for radio use. If I were specifying a battery for a station, I would go for the LiFePO like the Battle Born batteries -- better than the old Trojans, but $$ for sure.
For my low-current, Wifi and Pi station equipment, I'm using LiION and once I got it fully charged with the NOCO charger, I tested it and it ran for more than 3 hours on it's own -- powering two wifi routers, cable modem, two Raspberry Pi's, and some other low-current junk. That's more than I hoped for. I really only wanted the batteries to provide UPS service during a brief blackout or until an alternate power source could come online and I believe they also smooth out the ripple from the 60Hz-driven power supply -- wish I had a scope to verify that. The batteries I'm using are TalentCell, which I can recommend. I actually purchased them to run the LED's on a holiday light parade float (they've worked great for a few years now), but I thought I'd put them to use the other 51 weeks of the year, so they help power the Internet for the home QTH. I would also check out the TackLife 1200A jump-starter and (12800MaH) power source, or another power-brick that will do quick charging if your phone does that -- I use the 5V USB out to power the Pi's, but the TalentCell won't do the higher voltage USB for Quick Charge 3.0. Another good bet for station auxillary equipment would be a 12MaH LiFePO battery -- you'd just need a little buck converter if you have any 5V stuff.
The NOCO Genius 5 seems to be a quality product, but it is a bit costly for what it does. If it had more utility for automotive batteries, it might have greater value, but I found that it did not do anything my NAPA charging cart does not do, and the NAPA cart will charge 40A continuous -- it's got a softball-ball sized transformer in it.
The Foxsur's pulse feature might turn out to be valuable. It will take another week or so of treating a few old batteries I have and letting them rest to see if they hold the OCV.