I always pay with gc and utilize charge to room when I can.
TLDR for room charges visit the front desk night before check out and pay down the account with your gc. Have a CREDIT card on file for charging so pending charges or refunds aren’t holding up lots of cash in your bank account.
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Check-in: I do online check-in and enable room charging. If I have a balance due on my room Disney charges my cc on file morning of but I swing by the front desk at some point that day to pay it with a gc. Disney refunds the cc.
During trip: Usually I can’t pay the account down in the first couple days because the room payment sits there as a massive “credit” on the account that is used each day as the room rate bills to the account by day but this can be ymmv by CM. Sometimes the last couple nights I start swinging by the desk paying down the charges to prevent a large final charge/refund on my cc.
Check-out: Last thing before returning to the room for the night I go to the front desk and ask to pay all remaining charges with gc’s. Sometimes I forget to pay on the last night and Disney charges the balance to my cc. Nbd. I go to the front desk in the morning and it takes a touch of time too fix. Once in a while a CM will try to say “it’s complicated, we already charged your cc” LOL. But just be firm you want to pay with gc and they’ll end up refunding the cc the amount of gc’s you pay.
If we have a balance due on our room, I usually just call WDW a day or two before we go to pay it off so I don't have to deal with it once we are on vacation.
During the trip- some years ago, we were overcharged when charging to our room, so we have not used it in years. I also don't like how the credit card gets automatically charged the night before check out, and stopping by the front desk can be time consuming. I just had to do that last year to fix a billing error. Just one person was in line ahead of me, and it still took over 20minutes. Then as you say, the CM's sometimes are reluctant to reverse the charges.
We usually just use the giftcards as we go. When using E-giftcards, I just open them up on my phone. If using smaller cards, it is possible to combine them using the Disney giftcard website, but there are limits/rules, so it is good to read through them.
Always good to know, refunds go back to the Disney giftcard, so even if a card is used, it is often a good idea to hold onto them.
To keep track of physical cards, I usually wrap the most recent receipt around the plastic card. I also carry a mini sharpie (about 3 inches long). The mini ones are now hard to find, but a regular size sharpie also works. If a gift card is empty, I usually mark it with an X.
You can use giftcards directly to pay for G+, ILL's, and food mobile orders.
If we have a bunch of e-giftcards, it can get confusing to know which ones are fully used and which still have funds. I try to only open them as we go. When they are fully used, I mark those emails as 'important' and file them in a different folder. Again though, I don't delete them until later- just in case of any refunds.