There are no easy answers on who is the designated member.  You will not find any guidance in the law.  The only guidance can be found in Treasury FAQ U17.

A taxpayer that is a unitary business group must file a combined return under MCL 208.1511 using the tax year of the designated member. On the combined return, the business income of each member should reflect the accounting method the member used to compute its federal taxable income, whether cash or accrual.

"Designated member" means a member of a unitary business group that has nexus with Michigan under MCL 208.1200 and that will file the combined return required under section 511. If the member that owns or controls the other members of the unitary business group has nexus with Michigan, then that controlling member must be the designated member.

The combined return of the unitary business group must include each tax year of each member whose tax year ends with or within the tax year of the designated member.

The short answer is:  The "designated member" must have nexus in Michigan and should be the controlling member.  I have discussed with Treasury the problem with identifying a "designated member" where it is not clear who the controlling member is.  Treasury will be coming out with guidance shortly.  I hope they simply allow the unitary business group to pick a "designated member" with only one requirement, nexus in Michigan.  I also anticipate a form to be used to identify the unitary business group.