Estimating "how hot" the LM317 gets is a tedious calculation. There have been some OGR threads that have drilled deeper into the math. For your stated application of driving a 12V LED strip in a caboose or passenger car I suggest setting board's output current to 15 mA using an 82 ohm resistor. The LM317 case will be the hottest spot on the board but you should be able to touch it without burning your finger. Mounting the LM317 to double-sided foam-like tape will help as the tape will draw heat from the LM317 case much more efficiently that ambient air.
As an aside, for the same 15 mA output current, if you instead drive a single 3V LED (such as for a headlight) the LM317 will get hotter...and might be pushing the envelope but that's a separate discussion.
As long as the LM317 is of the SOT-223 package style, it will work. When you say "1 Amp" LM317 note that this board is intended for low-power applications...such as the 0.015 Amps suggested earlier. You will have serious heat problems operating this tiny board at 0.1 Amps or whatever.
Note that the 100uF capacitor might be marginal for flicker rejection. I believe GRJ's assembled board uses a 330uF - in other words 3x larger. Your mileage may vary. You might consider buying an assortment of lower cost thru-hole radial capacitors and soldering the leads to the SMD pads. This might provide additional mounting flexibility. Note that this bare-board can serve a variety of applications - including non-rolling-stock in which case a very small capacitor could be used so the entire board can fit in the cavity under Realtrax or Fastrack.
The MDB10SV package has 4mm lead spacing. The other two part number you show are ~5mm and ~2.5mm.