I think you're on the right track in this multi-season battle. I am going to give you a general write-up that I have below. I will add that you can treat it with herbicides any time that the temperatures are consistently above 50F...I usually think about this as when you're regularly having the mow the lawn. Fall is the best time for herbicides to translocate to the root/rhizome system...but that doesn't mean you should wait until then. Depending on the area and the product chosen you can likely treat it multiple time during the same year. Just be sure that you're realizing the full extent of control before re-treatment...usually 3-4 weeks and double check the label to see if there are limits on the number of applications or the quantity applied per year. 

I have included some information on an organization that may be able to offer assistance at the end

Please read this over and let me know if you have any follow up questions.
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Japanese knotweed (Fallopia japonica) is an aggressive perennial that reproduces both by rhizomes (underground stems) and by seed. It is often found in shaded to partially shaded habitats on moist, nutrient-rich soils, though it can thrive in gardens and on roadsides as well.

Physical control.
Repeated chopping or digging may eventually deplete underground reserves thus reducing the competitive edge of Japanese knotweed, however it is expected to take several growing seasons to achieve eradication. Both root and stem fragments can root and result in new plants, therefore any plant material removed for control should be placed in the trash headed to the landfill and not in yard waste or compost. Mowing is not recommended as it can spread the infestation. Mulching with most materials is not an effective means of suppressing Japanese knotweed. However, smothering it for several years with a thick plastic has been shown to be effective. Again, the practicality of this will depend on how large the area is, but you can read some detailed instructions from New Hampshire's Dept. of Agriculture at https://www.agriculture.nh.gov/publications-forms/documents/japanese-knotweed-control.pdf

Biological control.
At MSU Dr. Marianna Szucs laboratory group is working on a testing a biological control agent for Japanese knotweed, however this is still in the preliminary stages of testing for the state. More information Dr. Szucs program can be found at http://mariannaszucs.weebly.com/ in the Research section.

Chemical control.
Repeated herbicide applications of glyphosate or various brush killers, such as triclopyr (Garlon, Pathfinder, Brush-B-Gon) or triclopyr + 2,4-D (Crossbow) should help keep this plant under control, but eradication is difficult. Chemical applications can be made to cut stems for more effective treatment of small populations, see herbicide labels for details. Glyphosate (e.g. no longer in the store-available Roundup line of products, though there are generics available at stores like Tractor Supply Company, TSC, and farmer co-ops) is usually the best choice for residential users because of its safety, ease of application and lack of residual activity in the soil, however, in a lawn triclopyr or triclopyr + 2,4-D are better options because they will not harm the grass. Be sure to read and follow all labeled instructions. Repeat applications will likely be necessary. Also note that early fall applications (when temperatures are still above 50F) are often the best time to target herbaceous perennial weeds, like Japanese knotweed, due to better movement of the herbicides to the root/rhizome system with the natural fall movement of carbohydrates to those systems at the time.

There are more potent herbicides available that may provide eradication, but they are very persistent and can impact trees and shrubs with roots in the area. If you would like more information on these herbicides please contact me via email at <personal data hidden>.

**Note, if your neighbors also have this weed and they are not actively controlling it; it will continue to be a problem. I have seen that in my own yard.

Cooperative Invasive Species Management Areas (CISMAS).
In Michigan there are several groups working to eradicate invasive plant species. Each group covers a cluster of counties and is funded by varying sources, some volunteer efforts, and sometimes property owners themselves. Groups have varying priority species and their abilities to help vary. If they are not able to help with a particular species of interest they still may be able to point you toward local, reputable companies that can help. You can find your CISMA at this site: https://www.michigan.gov/invasives/take-action/local-resources Usually there is a coordinators email and phone number listed and sometimes there is also a web address (or you can do an internet search for the group’s name).

Mackinaw county is part of the Three Shores CISMA (https://www.threeshorescisma.org/). It might be worth contacting them as they have a form for Invasive Species Treatment Requests. They may be able to help you address this issue. The email listed on their website is <personal data hidden>
with a phone number of<personal data hidden>.