Capitalizing "the" when mid-sentence in a company name?

Solution 1:

The Guardian Style Guide gives the following:

[Lowercase] for newspapers (the Guardian), magazines (the New Statesman), pubs (the Coach and Horses), bands (the Black Eyed Peas, the Not Sensibles, the The), nicknames (the Hulk, the Red Baron), and sports grounds (the Oval).

[Uppercase] for books (The Lord of the Rings), films (The Matrix), poems (The Waste Land), television shows (The West Wing), and placenames (The Hague)

I think these things are convention more than anything else. Worth noting that they decide on the format 'The Lord of the Rings' rather than 'The Lord Of The Rings'.

Solution 2:

Strictly, this depends on whether the "The" is formally part of the company's name (e.g. its registered name).
If it is part of the formal name, then it should be capitalised as part of the proper noun.
If it is not part of the formal name, then it should not be capitalised.

[Your 'bonus' question is merely asking for opinions ("is it wise ...?") and is strictly off-topic on ELU.]

Solution 3:

I almost never capitalize the 'the' in something like your Association of Examples. It looks awkward, and it's really not even part of the name. For instance, you wouldn't refer to a medical study as "a the Centers for Disease Control study."

As far as branding is concerned, I think a capitalized 'The' is a subtle way to make your brand name stand out, but it carries with it a certain pretension, as well as a certain... conservatism, for lack of a better word. It seems to hearken back to the 19th century, somehow.