Does Barker take custody of user funds?

The deposit transaction goes from the user's wallet to your protocol's own contract. Barker is not a party to it, is not an intermediary address in the path, and cannot move it. If Barker went offline in the middle of a campaign, deposited funds would sit exactly where they already are and users would redeem through your protocol as usual — the campaign would stop, the money would not be affected.

What Barker does hold is the reward budget, and only for the duration of the campaign. It sits in an escrow contract and is released to a distributor on the published schedule; users claim from that distributor themselves rather than being paid out by us. Both contracts are on-chain and inspectable before you commit anything, so the failure modes are things you can look at rather than things you have to take on trust.