TL&DR
Using a larger carriage will, if too large, diminish the printing area.
To determine the correct values (for the X-axis) you:
- First need to assure that the nozzle can reach the origin, if not, try to see if you can move the endstop without the carriage hitting the frame, if not possible, your left margin will be decreased. If there was enough space for the carriage, and a value for
X_MIN_POS
defined, this value needs to be decreased. See How to center my prints on the build platform? (Re-calibrate homing offset) to set these margins correctly as you effectively need to reposition the print area on the build area.
- Second, define the bed dimensions temporarily to be the design size (do note to not bluntly jog the carriage to the end positions!).
- Now flash the firmware and home the X-axis, jog the carriage in increments to the right and observe the position on the LCD or through a connected application over USB. When the carriage is about to hit the frame, note the value, this is your bed X size (for a CoreXY make sure the carriage is clear over the whole Y-axis).
- Something similar can be done for the Y-axis.
- When you have noted the maximum sizes in X and Y direction, change the firmware and re-flash the printer board.
More details
The text below goes a little more in depth, first some definitions, then some theory.
Definitions
Axes
In reaction to:
The axis going from left to right
When facing your printer the left to right movement is your X-axis, back and forth the Y-axis and up and down the Z-axis.
-> Please note this is generally the most used printer convention, exceptions are applicable.
Dimensions
Hotend dimensions are needed for your slicer (when you need to sequentially print multiple objects), firmware uses bed dimensions and positions.
Bed dimensions
The following image reflects the changes to the build area when a larger sized carriage is used on the same printer (grey is the original size, red is the decreased area as a result of a larger carriage; note this is for fixed endstop positions!). Note, this is an example, CoreXY machines have limited space, so a decreased print area is more likely to be seen in both X and Y dimension. Prusa i3 style printers have the Y endstop generally on the bed carriage, so the bed size in Y will most likely not decrease. It totally depends on the construction of the printer, the extra space between the frame parts and the carriage (usually you can see that instantly by the endstop offsets, if there aren't homing endstop offsets, there is not extra space and the print dimension of that axis will decrease) or the ability to move the offset. Note that the bottom half of the image draws the out the result for losing X movement, the red area does account for losing Y movement as well, this is not uncommon for CoreXY, especially the printers with limited extra space for the carriage beyond the minimum and maximum bed positions.

How does this translate to your situation?
From Tiny Machines (TM) firmware (InsanityAutomation) you'll find that:
#define X_MIN_POS 0
#define Y_MIN_POS 0
This tells you that there is no offset from the homing endstop.
#if ENABLED(E3DHemera)
#define X_BED_SIZE 352
#define X_MAX_POS 352
#else
This tells you that there is less printing space in the X direction available.
From Marlin configurations (2.0.9.1), you can find something similar. It looks like somebody did a similar attempt, but added the offset to the X_MIN_POS
and mixed up the signs (it does hint to a 8 mm offset, so the creator probably also has a e3D Hemera extruder).
From the TM constants we read that the endstops in fact define the origin of the printer to be at the endstops (the MIN
values are zero) so the printer is at the origin when homed. Because of the larger extruder, already 8 mm of the X dimension is used, hence the smaller bed size of 306 - 8 = 352 mm. This is also the X_MAX_POS
, the carriage is not allowed further. You should now look into your custom mount and the endstop location and deduce your own "x mm" offset.