Do you need to use "as" with "brand" or "rebrand"?

The 'object complement' is well attested with such verbs as name, dub, call and crown, appoint:

• They named him 'Joseph.'

• They appointed her President.

Whether or not rebrand and brand enter into similar constructions is harder to show. I suspect that 'rebrand' is used more often this way, but neither usage is very common. Respectable-looking examples from the internet include:

The Commission for Equality and Human Rights, established under the Equality Act 2006, came into being on 1 October 2007. The Commission has since rebranded itself the Equality and Human Rights Commission (EHRC).

[Thomson Reuters Practical Law]

The British Association of Pharmaceutical Wholesalers (BAPW) has rebranded itself the Healthcare Distribution Association UK to reflect the evolving nature of the healthcare supply chain and the services provided by its members.

[The Pharmaceutical Journal]

So I'd say that the usage without 'as' is fairly rare with these verbs, but not unacceptable.

Examples including 'as' are more common; here is a typical example from the internet:

Radio Shack, for example, rebranded itself as "the Shack" in 2008.

[Wikipedia] ('Lost Market Share' paragraph)

Note that with a naming verb (and titles), quotes or italics were traditionally used, though this is becoming slightly old-fashioned:

• Company X rebranded itself the Smart Brand.
• Company X rebranded itself the Smart Brand.

That depends on what you mean, which can't be surmised since your capitalization and possibly punctuation is off.

Examples:

Company X rebranded itself The Smart Brand.

The above would mean that Company X changed its name to The Smart Brand.

Company X rebranded itself Smart Brand.

This would mean that Company X changed its name to Smart Brand. I mention this alternative in response to your lowercase T in "the" in your question's two examples. Company names are not like band names. If the company changed its name to Smart Brand, you would not put a "the" beforehand.

Take note that company names are NOT like band names. If a band changed its name to "Smart Band," it actually could be correct to say, "Band X rebranded itself the Smart Band," but only if the band choose to style the band's name with a preceding "the," for example, the Beatles chose to style the band's name with a preceding "the" that's not actually part of the band's name, but White Stripes, on the other hand, chose not to style the band's name with a peceding "the," even if some people sometimes do call White Stripes "the White Stripes," something Jack White, cofounder of White Stripes, has worked hard to quash. This convention, however, does not apply to company names.

Company X rebranded itself as the smart brand.

This would mean that Company X repositioned itself in the market as being the smart choice or possibly, if in tech, as being the company that puts smart technology in all its products (e.g., smart cars, smart phones, smart TVs, smart light switches, smart coffee makers, etc.).

Company X rebranded itself as "The smart brand."

This would that Company X has rebranded itself as such by using "The smart brand" as a slogan, which could also be written "The Smart Brand" since slogans can be written however companies wish to write them.

I offer the above examples because neither "Company X rebranded itself as the Smart Brand" nor "Company X rebranded itself the Smart Brand" is grammatical, the problem in both lying with "the Smart Brand," namely its capitalization, use of "the," and/or surrounding punctuation.