It's unlikely that Ohio either knew or cared if the Firelands militia conducted their drill on Friday or Saturday and their status would have remained unchanged. The state did not have an Inspector General to look at such things and, more importantly, the state did enforce the militia laws that were on the books. For example, Ohio's militia law required all white males between 18 and 45 to be part of the enrolled militia. If a person did not join, he was subject to a fine. But the state didn't enforce that law and never collected the fines. BTW, the money collected from the fines was supposed to purchase stands of colors for the regiments, drums and bugles for the companies, and camp equipage for the soldiers. Similarly, the state didn't enforce the requirement for every militia man to appear at muster with a firelock, pouch, shot, and powder. Nor did the state fine those members who failed to appear for drill. This also explains why, when war was declared, some townships asked the state to station a militia unit within their boundaries for protection. In other words, the state was pretty loosey-goosey (technical term) when it came to the militia and would have been happy just knowing the Firelands militia mustered at all.
In terms of who the militia reported to, it's quite likely that the Firelands militia did not belong to any regiment or brigade. Typically, in areas where there weren't enough people to form a regiment, the state authorized "Odd Battalions" in which 2 or 3 orphan companies were lumped together for administrative purposes. A Lieutenant Colonel Commandant was the commander of these battalions. My guess is that the Firelands militia was so far removed from the closest companies inPortage county that they were pretty much on their own. Many of the 4th Division, Ohio Militia, papers (and muster rolls) are in the Western Reserve Historical Society collection, and it may be possible to confirm/deny whether or not the Firelands militia belonged to an Odd Battalion of the 4th Division.
Speaking of muster rolls, it's difficult to recommend where to begin looking. Unfortunately, there's no single repository for them. Some are in the National Archives, some are in the Library of Congress, some are in state and local historical societies, some are in published county histories, and some are in private collections. Others have just disappeared ... if they were submitted at all. For example, it's not uncommon to come across letters in which a veteran is attempting to verify his militia service in order to claim his long overdue pay. That suggests that the appropriate muster roll was either not completed or was otherwise lost.
Ohio law allowed any state official to call the militia into service. Most of the time, Governor Meigs called out the militia, but there are instances where militia captains called out the militia for a week or two in response to some scare. If the state called out the militia, the muster rolls were sent through the Division to the Adjutant General in Chillicothe. However, if the federal government called upon the governor to provide a militia quota, the rolls were sent to both Chillicothe and Washington. For example, Secretary of War Eustis asked Meigs for 1600 militia to protect Detroit in April 1812. Ohio's Secretary of State alerted the state Adjutant General who allocated quotas for each Division: 1st Division, 7 companies; 2nd Division, 5 companies; 3rd Division, 5 companies; and 4th Division, 3 companies. Those rolls were going to be sent to both places, but the British captured all of Hull's papers, to include the muster rolls. Campbell's Portage county company finally settled its accounts with the War Department in 1814 for service in 1812, two years after Hull's surrender. Those units that did not participate in this First Army of Ohio mustered regularly for drill, but did not leave home until actually called into service.
I hope this helps answer some of your questions. Ohio's militia in the W1812 is, I think, a very interesting topic.