The short answer to your question is no, the warrant doesn't need to be equal to, or greater than, ALL the Delivery/Task orders issued (BPA calls don't even require a warrant). So please look at the following information which will explain the issue.
The first thing you need to understand is that no contracting officer warrant is needed to place "calls" against a Blanket Purchase Agreement (BPA). The contracting officer allows the "authorized caller" to place the call, however they are not given a warrant. They can only place BPA calls against BPA's where they are authorized to do so. BPA's fall under the procedures of FAR Part 13 and are a "simplified" method of acquiring goods and services. The "agreement" (BPA is not a contract action) becomes binding on both parties when the "call" is placed. BPA calls, like Purchase Orders, can be "refused" by the contractor since the BPA is not a contract. So the limit on BPA calls is the simplified acquisition threshold (or lower if specified), this can vary based on the situation, the "normal" SAT is $150,000. So the BPA limit is per call, not per BPA.
Your next contracting method you asked about is delivery and task orders. These are placed against "D" type contract actions, which include: 1. IDIQ (Indefinite Delivery Indefinite Quantity) 2. IDDQ (Indefinite Delivery Definite Quantity) and 3. Requirements Contracts. Delivery Orders is the term for "products" and Task Orders is for "services." A "D" type action is a contract, and IDIQ contracts have a minimum, your warrant would have to be equal to or greater than the minimum, since you are "obligating" the government to purchase the minimum. On an IDDQ action this would be no different than a "C" type contract action ("normal" one type contract), your warrant would need to be equal to or greater than the "face value" of the contract. However on a Requirements contract there is no dollar amount guaranteed, therefore the issue of warrant limits would apply to the "Delivery/Task Orders," which is also true of IDIQ and IDDQ contract orders.
So in summary no warrant required on BPA's limits are set forth in the BPA on who and what amount can be ordered under that agreement. For "D" type actions the KO dollar limit on the warrant is an issue for the minimum on IDIQ's, and the face value of IDDQ contracts and Requirements contracts don't specify a dollar amount, so the warrant would on factor in on Delivery/Task Orders placed on the Requirements contract (just like the other two D type contracts). I believe that you have a strong reason to contact your legal officer to see if there is some documentation showing that the Head of the Contracting Activity at NAVSEA has a mutual agreement with the Head of the Contracting ACtivity at NAVFAC "(for example") to share acquisition functions. Is is possible that they do share authority. Please see "1.601 General" where paragraph (b) says that some acquisition functions may be shared. If they don't share responsibility, then the head of one activity does not have the authority to sign the ratification of another activity.
(a) Unless specifically prohibited by another provision of law, authority and responsibility to contract for authorized supplies and services are vested in the agency head. The agency head may establish contracting activities and delegate broad authority to manage the agency’s contracting functions
to heads of such contracting activities. Contracts may be entered into and signed on behalf of the Government only by contracting officers. In some agencies, a relatively small number of high level officials are designated contracting officers solely by virtue of their positions. Contracting officers below the level of a head of a contracting activity shall be selected
and appointed under
(b) Agency heads may mutually agree to—
(1) Assign contracting functions and responsibilities from one agency to another; and
(2) Create joint or combined offices to exercise acquisition functions and responsibilities. 1.603.