What is the path to branch transfer to Logistics as a National Guard CPT?

There's a few steps in the process.
First, get the support of your chain of command. They probably want to know why, and can support you.
Second, get with the G1 shop in your state, via your S1 chain. Find out who can point you in the right direction for local procedures to follow and who to send things to.
Next, contact the 3 shop to enroll in training. You can take the Combined Logistics Captain Career Course via distance learning. You might not even need to go through the 3 shop for it. Then enroll in the resident phase.

Somewhere in this chain there may include a transfer to another unit. It's hard to justify sending an officer to a school outside his career field if that officer isn't in a logistics slot. Thems the brakes.

National Guard officers should cultivate a number of different career fields, to allow mobility.

Logistic's CCC seems to be a preferred choice among FA officers too. If you intend on a transfer, wouldn't you need a slot/ gaining unit?

This is a bit late, sir, but it's not very difficult at all (although it's quite arduous).

Basically, you are currently fedrec'ed as a 25A. You are eligible to attend any RCCC that you wish (which is the way to begin a branch transfer). Once you have completed the qualification course (or transition course), you need to re-apply for federal recognition in your new branch. Now, the only issue that has been occurring recently is that proponent (meaning the branch) apparently also has a say now. So if the Quartermaster Corps says there's too many LOG captains, why are we making another one, that could create some issues in the pipeline, but I generally don't see why in this particular case it would.