I'm sure Blair had a different perspective during his time as PM, but he ought to have known better by the end of 2008 that Putin had rather obvious imperialist desires.
By 2008, of course, Brown was in charge. Blair was very much from the Realpolitik school of thought, and like so many others he thought that the best way to get Russia to be peaceful (at least outside of terrorising their own citizens) was to ensure its economy was thoroughly enmeshed with other nations - the McDonald's school of peace if you will. In hindsight it has proved a failed plan but I'm not convinced that it was predictable that it would fail and, even if it was, it's not like the UK to decide the overall course of the Western approach to Russia.