If we ask the question another way -- "Did men from outside Indiana Territory fight in the Battle of Tippecanoe? -- then the answer seems to be YES.
The Wikipedia article Indiana in the War of 1812 says:
The native tribes who were part of the battle joined forces with the British in Canada at the onset of the war. Led by future U.S. President William Henry Harrison, American forces from Indiana, Ohio, and Kentucky were able to turn the tide of war on land that initially begun badly for the United States.
(emphasis mine)
The introductory section has no citations, however, so to get more information, you may have to do some digging. The article Battle of Tippecanoe, 1811 on the website of the U.S. Army Center Of Military History says:
Along with the 4th Infantry, by October Harrison had mustered twelve hundred men, including one detached company of the 7th Infantry, a company of the Rifle Regiment (armed at the time with muskets), Indiana militia, mounted riflemen, and one hundred twenty dragoons from Kentucky and Indiana.
You can find out more about the men from Kentucky via articles like John M. Trowbridge's Kentuckians at the Battle of Tippecanoe, published in Kentucky Ancestors (2006) V 41-3, pages 125-147. While this article may not answer your question directly, it has a bibliography which is full of ideas for further research. In particular, Trowbridge cites three works by historian Virgil D. White:
Thinking about the pension files in these indices led me to the US National Archives's website at https://www.archives.gov/. In Prologue Magazine's article Genealogical Records of the War of 1812 (Winter 1991, Vol. 23, No. 4), author Stuart L. Butler gives us an overview of the holdings at NARA. He says:
Compiled Military Service Records. The great majority of soldiers who served during the War of 1812 were volunteers, or members of state militia who were federalized for portions of the war period. There were also volunteer units directly raised by the federal government. The service records of these soldiers consist of compiled military service records or those records of service that were compiled from the original muster and pay rolls by the clerks in the Adjutant General's Office after the war (Records of the Adjutant General's Office, Record Group 94). The records are arranged by state or federal volunteer unit and thereunder alphabetically by name of soldier. A microfilmed index to these records is available on Index to Compiled Service Records of Volunteer Soldiers who Served During the War of 1812 (M602, 234 rolls).
So the question I have is this: was the Indiana Militia federalized, in which case, could there be pension files at the National Archives? (Note that the project to digitize the 1812 Pension files is now completely funded; the images are being placed on fold3 and are free to view.)
Were the soldiers awarded Bounty Land as part of their compensation being in this battle? If so there might be records in the Pension and Bounty Land Warrant Application Files, also discussed by Butler.
The CMSRs are compiled from rosters and have little genealogical information, just like the roster linked to in the question. To get to enough information to tell if this is your ancestor, or someone else with the same name, you need records with information in them, such as a pension application or bounty land records, or diaries of people who served in the militia, which might be in manuscript collections.
If the Indiana militia was not federalized, any records that survive would be held at the state level. Investigate all states where your ancestor might have been living at the time he applied for a pension.
The Indiana Archives and Records Administration's page on Early Military Collections describes their holdings as follows:
Indiana Territorial Militia, 1811- 1813
Taken from an early index and some of the muster rolls of the Battle of Tippecanoe and the War of 1812, the Indiana Territorial Militia index is available as a part of the Early Military Index on the Indiana Digital Archives. All information has been transcribed. Due to gaps in some records, not all soldiers, particularly privates and non-commissioned officers may be listed. Digital Images of the Indiana Muster, Pay, and Receipt Rolls, War of 1812 indexed by the Indiana State Library are also available on the IUPUI Program of Digital Scholarship Webpage.
The muster, pay, and receipt rolls will be subject to the limitations noted by Butler -- there won't be any genealogically relevant information for you. But what happens if you study the militia members as a group, instead of as individuals? Can you find them in the census, or in local histories? Did anyone else come from Ohio? Is there a migration pattern which is similar to the one you've traced for your ancestor?
If you can find manuscript collections with diaries, you might get a lucky dip and find someone who served in the militia with your ancestor (or his same-name counterpart) -- which might lead to more clues, which lead you to the information you want.
Study all parts of your ancestor's life. If he had land, work backwards in the deed records until you can find the original owner. (Hypothesis: Could the family story have come about because your ancestor or someone in his family purchased bounty land from a soldier who was in the battle? The FamilySearch Wiki says the bounty land awarded for the war of 1812 was in "Arkansas, Illinois, Michigan, and later, Missouri".) Then widen your search to his friends, neighbors and associates to see if you can find a group of people whose migration shows the same pattern as your ancestor's.
You may not find a single record which answers your question with direct evidence. It's very likely that you'll need indirect evidence and negative evidence to answer this question. Seek out posts from bloggers, local histories, and genealogical articles written by others whose ancestors fought in the battle, and see what resources they used and what references they cite. (See the lecture Inferential Genealogy by Dr. Tom Jones, CG, CGL, FASG, FUGA, for an example of how to use indirect and negative evidence to solve problems.)
Finally, one record type which is often overlooked when trying to fill out an ancestor's timeline are tax records. In his webinar "Using Tax Records for Genealogical Problem Solving", Dr. Michael D. Lacopo, DVM, reminds of the five Ws of journalism: Who, What, When, Where, and Why. Genealogists are pretty good at the Who (name), When (time period), and Where (geographical location), but we tend to fall down on the What and Why. He suggests checking state statute books to find out what was taxed and for what purpose the taxes were collected. Judy G. Russell's presentation "How Old Does He Have to Be?" addresses the same point with a wider variety of records.
This may require old-school "boots on the ground" research at county courthouses -- tax records were generated at the county level. But usually the taxes collected had to be sent to the state, so check state archives to see if they have state copies of the records.
In areas and times where records are scarce, gather everything you can find and analyze all the records as a group.
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