You'll find trickle chargers labelled as "motorcycle battery chargers" or occasionally as "maintainers" that are (according to claims) intended for long term connection to "maintain" a battery, like over the winter when you don't ride your cycle much. It's debatable whether they're really that much better than the regular auto chargers, but they're smaller and should do the job for you.
The "maintainer" style aren't much good for bringing a low battery up, since they generally show about 2 amp maximum charging capacity. They'll get the job done, but could take several days to do an initial charge on even a decent sized motorcycle battery. (Lots of those cycle batteries come "dry" and you have to add the water and charge them before using.)
Many of the little ones have screw holes in a flange so that you can mount them on the wall near where you keep the bike (or on the bike?), and no "carry handle," and some don't even have a light to tell you if they're on/connected. Fancier ones, not much different in price, may have two or three status lights that tell you 1. if it's plugged in, 2. if the battery is connected, 3. if the battery's connected reverse polarity, and maybe one or two other functions.
The decent "auto battery chargers" often allow you to select 6 or 12 volts, show a maximum charging capacity of 8 to 12 amps or so, often have a "charging rate" ammeter to show how much is coming out at any given time, and may have a "hot shot" setting for "starting" a car with a very low battery that may put out anywhere from 30 to 80 amps. Most of them will claim a "trickle charge" ability for keeping a battery up, and "intelligence" that decides automatically when to taper down to that. One in this type likely will run closer to $40 to $60, although "bargains" sometimes do pop up in the ads.
For long-term "charge maintenance" most battery makers recommend a "cycled" charge where the maintainer senses battery voltage and only applies its own when the battery drops off a little, or a "timed cycle" that applies a little higher voltage for a while and then lets the battery "settle" for a different time. Most "not-very-intelligent" chargers/maintainers just drop the charger voltage back to around 12.8 to 13.4 V (from a higher voltage during charging) when the current the battery is taking drops to 1 or 2 amps.
An ammeter is pretty much useless during the maintenance trickle, since the current level is too low to register on the ones commonly used. It's of some use when you're charging a low battery since you can watch the needle drop and guess how much you've accomplished. Most newer designs just use LEDs for "status indicators" and the new ones without a meter may do a better job, although the "specs" and "manual" that come with any of them almost never tell you what you need to know to tell how good a given one is.
As to putting the battery on a board, manufacturers all say it's unnecessary. Old batteries had cases made of pitch/tar that could discharge some without an insulator (board) under them, and some old ones "spit acid" or "ooze acid vapor" and wood is more resistant to the acid than most other things that might be around. The board is not necessary with any modern battery for protecting the battery from discharge, although having a little air circulation under the battery will minimize condensation and dampness that can degrade concrete.
A better solution than the board would be a real "battery box" like many boats (and campers) use, but it may be impossible to find one small enough to be really suitable for a keep-alive battery of the size you really need. If you add an external battery as a replacement for the one in the welder you can use almost anything that's "big enough," and you might choose one big enough to fit into one of the battery box sizes generally available. "Too big" a battery probably won't hurt anything, although a bigger size (within reason) won't really affect how long one lasts by very much and the price does go up for bigger ones in the same style.