DISCLAIMER: This is not final, but could already work for you. I suggest to print the testpiece first (-:

General features:
Fully configurable / customisable with attached OpenSCAD file. Studs are chamfered, tightness of fit adjustable by customizing. 

Special feature:
The 16x12 version provides a Duplo compatible base plate that uses the full length of a Prusa MK3S-sized print bed, and should - while having slightly shortened borders in the X-dimension - be fully usable.

Basic parameters:

• Number of the studs in X and Y direction

• Diameter of the studs, (tune this if parts fit too loose or too tight)

You could also change, but probably will not need to:

• height of the studs

• corner roundness of the plate

• size of the chamfer in the top of the studs

Special parameter:
If you want to maximize the number of studs you can print on a MK3S-sized print bed, you can also

• shorten the outer borders in X direction by a given amount (preconfigured in the attached 16x12 files)

Pretesting suggested (once again ;-)
To check, whether the big preconfigured STLs oder .3mf would work for you, you should first print the TESTPIECE. Stud dimension of the provided print files give a good but not too firm grip on my MK3S+.

FILAMENT AND PRINT PARAMETERS
You probably can see my print settings in the attached .3mf-Files. On my MKS3+ with a 0.4 mm nozzle, I used 3 perimeters with a perimeter width of 0.6 mm (yes, 0.6 mm), 30% gyroid infill and a layer height of 0.15 mm. Seems to work for me.

I printed the testpiece and the shortened 16x12 version in PETG. I did not test the 14x12 version.

The stringing in the pictures of the green and red prints is probably caused by using cheap old PETG for my test. The yellow prusament PETG testpiece had no stringing at al.

Attached configurable files:
If I did not loose track of my testing, all attached OpenSCAD files only differ in the given number of studs in X and Y direction, and in case of the 16x12 files the outer borders in X-Direction shortened by 3 mm.

How to configure/customize:
If you load the .scad-files in OpenSCAD, you should be able the parameters in the source code, or to open the paramenter section in the built-in customizer. After reconfiguring press F5 the recalculate the preview in OpenSCAD. If everything looks correct press F6 for calculating the final object. For a large base plate, this can take a couple of minutes. After that step, press F7 to export the object.

Why not Lego compatible version (yet)?
Yes, changing most parameters to half the size should work out to get a standard Lego compatible base plate. But I will have to test and fine-tune that later. Especially for a snug fit.