Bit late to the party, but the answer to the thread title is a resounding "yes you should".
I think some forum members, and even rail passengers have become very desensitised to what Delay Repay actually is. You are making a formal claim for compensation, albeit under "industry arrangements". The onus is on you to support your claim for compensation. Delay Repay is one of three methods proscribed in the conditions of travel for you to exercise various consumer and contractual rights when delays occur. In essence, Delay Repay is essentially a reverse "out of court settlement". The TOC pays you compensation, you agree to accept it as a remedy, and the matter does not need to be escalated further. You are not obliged to use a Delay Repay scheme, and the conditions make it clear you have 2 other options, 1 of which takes you down the path of exercising your statutory rights (which could eventually end up in a civil court).
So I am afraid whilst you are not legally obliged to retain records, given that from a civil perspective (in England & Wales) you can be sued for up to 6 years afterwards, prosecuted in the Magistrates' Court up to 6 months afterwards, and in the Crown Court at an indefinite date long into the future, it would be careless not to. Simply because "Delay Repay" has a "casual" feel to it, does not take away that you are presenting a formal, legal demand for compensation/redress each and every time you claim, and affirming that it is true and accurate.
99% of the time, I would bet that you would never be contacted about that claim ever again. Unless I am missing something, it appears that the actual number of people receiving letters about "potentially suspicious" claims is likely somewhere around 1% or maybe even lower, which to me, suggests that whatever processes are currently in place for the rail industry are generally working remarkably well for the vast majority of claimants.