Your two points certainly merit consideration by OP. I do rather complex things in my Mac-intensive work, including moving between laptops and desktops. In all of that, I can't really think of a situation where application state that could be preserved by a battery-backed Mac would be an issue subbing in a non-battery Mac. Basically, I would "save all", change locations, "open" and get back to work again. I do use a couple of non-battery desktops to get some things done and if they are to work on the same stuff, I move the files Mac to Mac in iCloud, Dropbox and/or on portable drives. No big deal... and no preserving states between them as I'm fully closing them on one to then open them on another.
However, that shared, there probably is some things where fully saving, closing, unplugging, moving, plugging, opening (at other location) and resuming the work MAY be an issue. If so, then the idea doesn't fit the need.
As to "secondly", that is a weakness of all Macs. It's too bad that Mac can't dynamically scale the UI to maximize whatever resolution of screen that is attached (as Windows can). However, OP says "the monitor at work" so the suggestion replicates that. While possible the monitor at work is a 5K one, my guess is that it's simply much larger than the 13" one on the laptop OP uses. In a "bad eyes" scenario, size of screen will certainly help even if NOT 5K. Conceptually, OP could do most of their work at work and use a big screen 4K TV for a little spare work at home.
In my own experience, when I've had to attach Macs to projectors or screens in meeting rooms, NONE have been 5K (or higher) but all have been BIG screens. MacOS looks quite good at 4K or even 1080p if the screen is big. Of course, it is not as sharp as 5K but readability is different than pixel sharp scaling to macOS targets. My guess is big screen at OPs work is not 5K, so big is trumping 5K for OPs eyes. If so, similarly big (screen) at home may do the trick.
Given the mass work-from-home stuff we've all been through last few years, lots of work-purchased tech is coming back into offices as people return to offices. My suggestion of chatting with I.T. Department for a spare screen that is the same as the work one probably has real potential. Many I.T. departments I encounter are piling up equipment purchased in support of work-from-home. I suspect there are going to be I.T. "garage sales" galore to blow out some of this returning tech at companies all over the country... and/or sites like Ebay are going to be swamped in much more used, not-too-old tech being dumped to clear all this out of company storage with much less work-at-home need.