This route assumes the game is run in DOSBox at high cycles (or natively on a fast PC), allowing zipping at Fastest speed. Prepare the first command, "open door" (or "open closet"), and the ideal menu position (selecting Fastest speed) before restarting the game.
Commands are given in uppercase for clarity, but need not be typed in uppercase. Fastest speed should be selected by pressing Esc followed by Enter.
[prepare menu position and "open door"] [restart game, start time] [Fastest] 948716898927143868 OPEN DOOR TUR[F3] 27161421218923672[Enter]6121 [F10] (navigate around the turn, preparing at least part of "look map" as you go, and head east. The location where you need to go east is pixel-perfect. If you are 1 pixel too far south, you can correct with 936 after the first zip. If you are 1 pixel too far north, you die.) 6 [Fastest] 134 [F10] LOOK MAP [Enter, Enter, F8, Enter] (navigate to some way behind the rock, 2, Fastest, F10, navigate east and around the turn. Go west until you think you can reach the next screen by zipping southwest. Press F3 during the walk) 1 [Fastest] 4 LOOK MAP [Enter, Enter, F6, Enter] The next sequence depends on which of three random positions you get dropped in. If you appear in the western part of the screen, follow with 2936868 OPEN DOOR. If you appear in the middle, follow with 2926868 OPEN DOOR. If you appear in the east, follow with 634245 OPEN DOOR. The eastern RNG is preferable; not only does it save one keystroke, it also avoids an RNG encounter with the bears.
Finish by preparing "PET DOG", then 417 and press Enter at least twice. (Note that the storekeeper will greet you shortly after you enter the shop. With perfect timing you can pet the dog before being greeted; but if you are slow or unlucky with the framerule, you'll have to dispel the greeting with the first Enter before finishing the run with the second Enter.)
In case of bad bear RNG, dismiss the bear message, hit the previous direction again, and then continue the expected sequence; e.g. 293[Enter]36868 OPEN DOOR 417 PET DOG.