[cherrypick][refactor] additional error login in faucet (#9987) (#9988)

## Description 

Adding some additional logging in faucet to get visibility in errors
produced probably via some middleware, as the only thing we currently
see is:
```
sui_faucet::metrics_layer: Request failed in 0.00s
```

## Test Plan 

How did you test the new or updated feature?

---
If your changes are not user-facing and not a breaking change, you can
skip the following section. Otherwise, please indicate what changed, and
then add to the Release Notes section as highlighted during the release
process.

### Type of Change (Check all that apply)

- [ ] user-visible impact
- [ ] breaking change for a client SDKs
- [ ] breaking change for FNs (FN binary must upgrade)
- [ ] breaking change for validators or node operators (must upgrade
binaries)
- [ ] breaking change for on-chain data layout
- [ ] necessitate either a data wipe or data migration

### Release notes

## Description 

Describe the changes or additions included in this PR.

## Test Plan 

How did you test the new or updated feature?

---
If your changes are not user-facing and not a breaking change, you can
skip the following section. Otherwise, please indicate what changed, and
then add to the Release Notes section as highlighted during the release
process.

### Type of Change (Check all that apply)

- [ ] user-visible impact
- [ ] breaking change for a client SDKs
- [ ] breaking change for FNs (FN binary must upgrade)
- [ ] breaking change for validators or node operators (must upgrade
binaries)
- [ ] breaking change for on-chain data layout
- [ ] necessitate either a data wipe or data migration

### Release notes

Co-authored-by: Anastasios Kichidis <akihidis@gmail.com>