Unless I am mistaken, Moynihan first sat in 1997 as a hereditary peer, 2 years prior to the reforms under the House of Lords Act, after which his status was changed to that of a life peer.
Like it or not, Moynihan's current status as a life peer is predicated entirely on his prior, and intrinsic, status as a hereditary peer. His change of status does nothing to change his basis for sitting in the house--he was not appointed for "service to his country in his field".
Getting to the substance of your post, though, even if that was Moynihan's exact contention as you state, it is incomplete and selective in its perspective. The "right to choose athletes" is not an unqualified right, and is subject to agreed-upon conditions imposed by governing bodies, incorporated into domestic law by treaty.
You suggest that the Brits "should be able to enforce that criterion", but you assume what you should instead prove. The point of the CAS proceeding is to determine that very issue, on conditions to which the Brits agreed, and by which the BOA is bound.
The only thing about which Moynihan is "right" is about expressing the sentiments of some people with a view on the issue, which people include athletes. British athletes do not all have the same view on this issue, however, and therefore Moynihan is wrong when he purports to represent the interests of "athletes" in general: "“I’ll be disappointed if we didn’t win because we fought hard to represent the interests of the athletes,”
Chambers is himself one of those "athletes" whose interests Moynihan purports to represent.
I agree with you about the difficulty of letting convicted dopers compete again, but at some point, one must acquiesce to the decisions of a larger society and get on with life, while maybe working to change the landscape.
And who knows? The CAS might ultimately decide in favour of BOA. Don't jump out the window just yet.