The problem with this POD is that it's not a matter of Harrison missing Tecumseh, but Tecumseh missing Harrison. (I have a book in front of me: 1812, The War That Forged A Nation, as a definite source).
In September of 1809, Harrison engineered the Treaty of Fort Wayne, where the NAs (Native Americans) essentially gave up half of Ohio and Indiana for settlement. The Shawnees were currently living on that land, but had only recently migrated there, and Harrison believed the Miami NAs had a stronger claim. This bargain will definitely happen in any TL, because Harrison was pursuing land any way possible.
At the time, Tecumseh was in the south, with the Creek NAs, trying to rally them to his alliance. When he heard of the deal, he denounced Harrison's land grab and vowed to keep the land unsettled and unsurveyed. He would do this in any TL as part of his personality and vision.
Harrison heard of this (also in any TL), and invited Tecumseh to come and present any claim to the lands possible. So, Tecumseh came, as well as 400 warriors. They met twice, the second time in the summer of 1811. Tecumseh exchanged only rhetoric, pursuing the larger goal of his alliance, of which he may have boasted too openly. He vowed to return and talk again in the Spring of 1812, but asked that, in the meantime, Harrison not make any surveys or settlements.
And, in OTL, while Tecumseh was away, Harrison took 900 men, including the 4th Regulars, and some Indiana and Kentucky militia. He reached Prophetstown on the lands in question on November 5th. During the night of the 5th-6th, The Prophet led the attack against Harrison, and despite initial surprise ultimately was totally beaten. So Prophetstown was burned.
A workable PoD for this is that Tecumseh gives The Prophet explicit orders not to attack Harrison's force, even should opportunity allow, unless as a necessary defensive measure, and instead try to exchange rhetoric.
History will never know if Harrison would have attacked Prophetstown anyway. Most sources seem to say he would. However:
1) On the off chance that he wouldn't, there is no way in hell that The Prophet would voluntarily withdraw, so the town would remain until Tecumseh's return, and he in turn would gain prestige and respectability by refraining from ambushing an American column on an "entirely peaceful" mission.
2) More likely, Harrison would try to conduct an attack. His force was too small to conduct a seige and he had no artillery to speak of. However, given time to construct someting of a semi-proper defense, plus fighting purely on the defensive, I'd give the NAs a better chance to win the fight, given relatively equal numbers, which they had. It would be bloody, and Harrison's force would not be routed, but it might be forced to retreat.
In either case, Prophetstown, and so Tecumseh's alliance, lives to fight another day.
On the other mini-topic: No, the Americans would suspect the British of supplying the NAs with firearms no matter what, they already had before Prophetstown and the trent was towards animosity (although nominally the main reason for fighting the war was impressment).
In the War of 1812, it will procede as in OTL until Summer of 1813. Even after Perry wins control of Lake Erie, Harrison will be leery of advancing all the way to Detroit, because his left flank will constantly be menaced by Tecumseh and his warriors. This probably means no Battle of the Thames, so Tecumseh lives even longer.
Also, affairs in the South might go more favorably towards Tecumseh, and so maybe he will enlist the help of more than just half of the Creek NAs, in which case Jackson cannot have as great a victory as in OTL.