What is the opposite of "strategic"?

"Non-strategic" is broadly used to mean the reverse of strategic as defined in Wiktionary:

Not strategic; not related to strategy.

It is a waste of military resources to bomb nonstrategic targets.

Regular might be a good candidate, however, it doesn't have such a connotation of not strategically important. Strategic means in Merriam-Webster:

useful or important in achieving a plan or strategy

In that sense, I think ordinary works better than regular.

Having no special characteristics or function

The usual antipode to strategic is tactical

A strategy is a larger, overall plan that can comprise several tactics, which are smaller, focused, less impactful plans that are part of the overall plan. While the original usage of the terms strategy and tactic was in a military context, they are now used in a wide variety of everyday settings, including business.

diffen.com

In simple language, strategy is big picture/long-term, while tactics are focused/short-term.

The accounts are strategic, not the account manager.

• The Strategic Account Manager maintains and expands relationships with strategically important large customers. Assigned to three to five named customers, the Strategic Account Manager is responsible for achieving sales quota and assigned strategic account objectives. The Strategic Account Manager represents the entire range of company products and services to assigned customers, while leading the customer account planning cycle and ensuring assigned customers’ needs and expectations are met by the company. The Strategic Account Manager reports to the Vice President of Strategic Accounts.

That said, to answer your question, I would call accounts that are not strategic, "regular accounts" or "regular savings accounts" and the professional "Regular Account Manager".

• Addendum (credit to chasly from UK): according to the blog InsideView,

  There are two types of accounts, regular and strategic. Both are important and need to be given the time and resources to be successful but strategic accounts has a larger impact on the company and will get some extra attention. The important thing to remember and be disciplined about is that NOT every account is a strategic account. The most important consideration for what makes a strategic account is whether the account you are working with desires a lasting win-win relationship with your company.