Ok - before you start drilling holes here, let me interject.
I work as technical support in a large industrial supply company for machining customers, and have been making parts since the early 90s.
First thing you should consider it getting a way to hold the surface you want to drill square and locked solid so it doesn't move because you will need three tools to make the hole. A good spot drill, a good drill, and then the reamer. Don't just throw a drill in a DeWalt and go to town, I would suggest a drill press at least and some kind of jig / fixate to hold it solid and still while you change tools.
Second, when you drill thru the carrier and into the ring gear, you might have issues. Ring gears are heat treated high alloy steel, usually in the upper 50s on a Rockwell C Hardness scale. Hardware store drill bits are not going to do this, they are around 60-63HRC. Your gear is probably 57-59HPC, so basically the drill and gear are the same hardness. You will not make one hole.
You will need cobalt at least, a way to cool and lube everything while cutting, and you will need to go slow.
I would highly suggest a spot drill - this will ensure a strait hole. Cobalt, 5/16" or larger -
https://www.mscdirect.com/product/details/58816141
If you have a good solid setup that won't let the part move, you can use a 6.1mm drill - which is only about .009" smaller than .250". However if things are moving you will end up with a "tri-lobe" hole that will be bigger than .250"
For a drill -
https://www.mscdirect.com/product/details/80994221
Run the drill no faster than 500rpm, or you will burn it up in steel that hard. Use oil.
For a reamer -
https://www.mscdirect.com/product/details/30145577
Run the reamer no faster than 380rpm. Use oil. Any faster and you will end up welding the reamer in the hole.
In my god honest opinion - dowling the gear to the carrier is not going to do much if anything, with the surface area and I assume a press fit (the ring gear was probably heated to go on the carrier), it should be the last thing in that differential that moves. It would be as if both pieces were made of one.