Why wasn't he sent in some capacity? That might be a way to approach the question.
He was young enough, active enough, prominent enough, and apparently in good with the government. However, IIRC, he asked for a posting to the Peninsular and was turned down. He didn't have in any active military role after the Revolution.
So, why didn't the military establishment use him after the ARW?
If you can determine why they didn't use him, you might have a good handle on how he'd handle himself during the war.
Actually, they did use him, or considered doing so. He was almost sent to India with Cornwallis to command the cavalry there in the years after the Revolution, but his politics (he was a Whig, and the government at the time was a Tory government) caused the appointment to be rescinded. He was put in command of British defense forces in southeastern Ireland in 1802 when it looked like Napoleon was going to try an invasion, and he was assigned to organize defenses in southwestern England when it looked like Nappy might invade in 1805.
As for why he didn't get the assignment to Spain which he had wanted, it seems, again, to have been due to his politics. He was originally a Whig, but shortly before the death of Pitt the Younger in 1806, with whom he had become friends, he had defected to the Tories. But, his own political opinions were always more aligned with the Whigs than the Tories, and he tried (unsuccessfully) to get back into the Whig Party. By that time, the ruling party was the Tories (after a brief interlude of Whig rule after Pitt's death), and thus, by going back to the Whigs, he cut himself off from both parties right at the time when the assignment to Spain might have been had.
So a POD might be that Tarleton remains loyal to the Whigs. As in OTL, Tarleton campaigns against Wellington, trying to get command of the Peninsular Army. However, Wellington still gets the assignment to Spain in 1809 as per OTL. A bitter Tarleton spends his time castigating Wellington in Parliament, until the fortuitous declaration of war by the United States is received. To shut Tarleton up and reward a loyal Tory, the Tory government promotes him to full General and grants him the command of His Majesty's forces in Canada. Tarleton accepts, and departs for America in July 1812, arriving there before August 1 of that year.