Morning back at you
Great you have confirmed the "casting date code" to end in a "3".
That helps confirm it's a '53 model year unit. To be specific, it decodes as:
H = August
27 = 27th day of August
3 = 1953
The second number you provided "3686832 GM" isn't a "serial number". It's actual the casting/part number for the transmission shifter side plate unit.
According to the Tech Service Bulletin (TSB) # 249 (found at the beginning of the above linked 1953 Chevrolet Info Kit) the serial number for your transmission should be found on the small machined flat at the upper rear section of the main case, just behind the edge of the side shifter cover. See a screen grab of your original pic with an arrow pointing to the spot where the serial number should be.
More details from the TSB# 249 show the serial number should (for your 3 speed) begin with the letter "M", followed by 1 or 2 numbers to indicate the month and finally 2 numbers to indicate the day of the month (see the other attached screenshot).
Since your main case casting number and date already confirm you have a 1953 3 speed, up to you if you want to go further to try and clean that machined pad and verify by serial number what you have.
Let us know what you decide/figure out.
Dan
