I have replaced hundreds of panes in bookcases and corner cabinets over the years, this is how I do it.
Take the door off, lay it on the workbench and support the glazing bars on the underside.
To remove the old putty I first try to chisel it out with a sharp chisel, sometimes it will practically fall out but usually not. The next step is to heat it up with a fishtail attachment on a heat gun. Keep the gun moving over a six inch section then chisel it out. It will mostly peel off easily at this stage unless the glazier has added plaster of Paris to the original putty!
Carry on round until all of the putty is out. Lift out the remaining glass then heat up the the bead of putty that was under the glass and chisel out. Be carful not to use too much pressure as the beading will be very delicate at this stage. When the recess is nice and clean you are ready to re-glaze.
Get a small tub of brown putty, take all of the putty out of the tub, massage it well then replace back in the tub until you are ready to use it. I usually buy white putty and add dry earth pigment to match the colour of the old putty but I don’t think that is necessary for your cabinet.
You can get 2mm glass from your local glazier, they won’t be keen on selling to you but just tell them it is for an antique and they should be fine. You could also use greenhouse glass but this can be a bit heavy for a cabinet this size. Place the glass over the aperture and trace round the shape with a sharpie. Cut the glass to size.
Take a small handful of putty and push it into the bead, place the glass on top and with even pressure push the glass down making sure the putty oozes out evenly. Scape off the excess putty. A few very fine pins can be knocked in to hold the glass in place but is not always necessary. Push putty around the bead and finish off with a good clean putty knife.
The putty can take up to three weeks to set which is why some restorers add plaster of Paris or even hide glue to speed up the process, I tend to work in some terebine driers when mixing with earth pigments.
Hope this helps.