First let me address the lateral adjuster: if this is just going to be a user plane and you can’t use the laterial adjuster because of the short length you can always do the adjustment with small hammer taps just like doing it for a wooden plane. I have never had to do it, but I have read that you can carefully file or grind the lateral adjustment post/pin end at the bottom and remove the adjuster. Then do the same on a parts plane and put that adjuster on the old frog. The adjuster will still be loose on the frog but it will stay in place and do it’s work a long as the blade is in place. I’m sure some of the more metal work savvy forum users could provide some technical suggestions for permanently affixing the adjuster.
By the way, in my opinion, unless you really just like going through the metal plane flattening process for fun, with a few exceptions, the current fad of bed flattening should really be reserved for smooth planes, and to a lesser degree, jointer planes and block planes. Chris Schwarz has some video out that can guide you when deciding which planes really need to be super flat and which ones need to be “sorta flat”.
If you have any ground metal surfaces in your shop such as a jointer bed or a cast iron table saw table you can use them as support surfaces for your glass and as a check for your wooden plane sole flatness. I also have used marble slabs and tile but not before checking them with straight edges. I do my wooden sole work with all of the metal removed. It makes it easier to hold. Some have suggested you need to have the plane all together and under pressure but I haven’t had a problem doing it the way I do it. When flattening wooden plane soles I use a jointer plane with little or no camber. I check the sole with a good square and straight edge to get a general idea of square-ness to the edges and the mouth opening and the flatness condition. If the sole is fairly close to being in good relationship to those surfaces I then turn my attention to flatness. If the sole is badly out in relationship to the mentioned features the you have to address that along with the flatness, generally square everything up and then concentrate on the flatness of the sole. flattening the sole is the same as with any other piece of lumber (lots of videos online for that), repeatedly checking for flatness and wind. As an extra step, once you feel that you have the surface flat you can take the sole, if the tested, ground surfaces mentioned above, are available and check to see if there is any rocking. If there is no rocking then the plane is flat enough. If there still is some rocking try and determine if it is front-to-back (convex along length), side-to-side (rounded across the face), or corner-to-corner (wind). Using extremely light cuts you can remove very thin shavings from the high spots until the plane no longer rocks. It is possible to create a concave surface that wouldn’t rock but most people are inclined to plane convex surfaces and those normally show up with the straight edges. I then round the edges slightly if necessary to reduce splitting, splinters, and ease use . Generally I just apply a little BLO, and paste wax before I assemble and adjust the tool. I might have to re-mouth the plane afterword depending on how I am planning to use it but that is another subject.