Does anyone know if it is permissible for a Pension Administrator to charge their administrative fee as a percentage of plan assets, (i.e. basis points on plan assets) or must it be a flat or hourly fee per plan or participant? Assume the TPA is not offering any investment advice.

Permissible? Fees are ususally set by competition and negotiation between the parties.

I have seen many that charge basis points on assets under administration. The calculation must be detailed in the administration agreement.

Since this thread is under Securities Law Aspects, let me raise some securities law issues. The bad news: (1) such an arrangement may require a securities license (either Series 6 or Series 63/65) under state securities laws to be legal; (2) some states define a "profits interest" as a security (requiring registration). This means a percentage of the profits, so that fees should be based on all assets and not on earnings; (3) ERISA does not pre-empt state securities laws.
The good news: (1) an offering to a qualified plan is likely exempt under state securities laws. However, I suggest reviewing state securities with a securities attorney.