No, this is not possible to relax this policy. unsafe-inline is specifically ignored by Chrome Extensions since manifest version 2.
Documentation (emphasis mine):
There is no mechanism for relaxing the restriction against executing inline JavaScript. In particular, setting a script policy that includes ‘unsafe-inline’ will have no effect.
The error message mentions several possible ways, but the docs are clear that no CSP will allow inline scripting, and ignoring unsafe-inline is but one of the measures.
Update
As of Chrome 46, inline scripts can be whitelisted by specifying the base64-encoded hash of the source code in the policy. This hash must be prefixed by the used hash algorithm (sha256, sha384 or sha512). See Hash usage for elements for an example.
See this answer for more in-depth look at whitelisting.