Having never used a handheld planer for this I think you can do this the same way you'd do it with a hand plane, if it can be adjusted to take a light enough cut.
You're simply planing the board more towards one end, in stages:
- plane first from about 3/4 of the way along the board
- next couple of passes from about halfway along the board
- last passes to get the edge nearly flush over approx. the last quarter of the board
- one or two very very light final passes along nearly the whole length to flatten out
I do a lot of tapering by hand this way on a small scale and the process does translate to longer surfaces such as board edges, although it's more difficult when working larger to get a flat edge after you're done.
Which direction you plane the board to prevent tearout is an additional uncertainty. Most wood has a preferred planing direction, and if on your board edge that's away from the lip then you'll have to work opposite to the above outlined method, planing towards the 3/4 mark, then the halfway point etc. This will involve lifting the planer away from the wood as it runs, and I don't know how easy or safe that is.
In either case I think you should protect the edge of the wood at 90° to the board you're planing with a couple of layers of tape.
Once the uppermost layer of tape is getting chewed by the planer then it's time to stop planing and switch to something else. I would take the last of it off with a file, then do finish scraping or sanding, but you could of course skip the file and do all the work with sandpaper, it'll just take longer. In case it needs to be said: back the paper with a block.
Or is there a better tool for this job that's relatively inexpensive?
I'm presuming here that you don't have a belt sander, which is the tool I would use for this if I didn't own a hand plane.
If you wanted to buy something specifically for this I'd get a low-angle block plane. One of these has a lot of potential future utility, so it's a very worthwhile investment (lifetime tool too, won't ever wear out and has no consumables like a sander). But, unless you buy high end you will need to sharpen it for it to work successfully first time.