NOTE: Apparently, BitPay is enforcing some kind of daily limit on GNOME, probably for KYC/AML regulatory reasons, so you may have your donation denied.
The goal is to withdrawal to a donation address a certain amount of Bitcoin from ChangeTip. To be precise in your numbers, you must tell ChangeTip to deal in BTC rather than some other local currency.
To do this, go to this page:
Set your preferred currency to "Bitcoin".
To begin donation, go to this page:
In the Bitcoin donation form, just fill out the amount you want to donate and select the currency unit (e.g., USD, BTC, etc.).
Click the big "donate now" image button, and BitPay will generate a bitcoin address to which you may make the donation.
There’s a countdown; presumably you have only that long to make the transaction before the process times out, which seems somewhat silly.
Anyway, copy the textual version of the bitcoin address.
Open up your ChangeTip "Money" page:
On that page, you’ll find a withdrawal form, where you can input both the donation address and the amount.
BE CAREFUL: BitPay tells you how much BTC to pay, while ChangeTip may ask you how much mBTC to pay; it may depend on your settings.
ChangeTip makes noise about needing to verify the withdrawal, stating that it could take hours; fortunately, for me, BitPay’s page acknowledged the payment almost immediately, and shortly thereafter blockchain.info was able to show my transaction, to which ChangeTip linked via the "Withdrawal History" on the Your Money page; ChangeTip paid the transaction fee.
It is worth noting that by January 15, 2015, ChangeTip will begin charging a 1% withdrawal tax, so this method may not be desirable in the future.