Using "a tiny" in the same way as "a little"

This is similar to what Matt Эллен said, but uses the analysis and terminology of the Cambridge Grammar of the English Language. First, tiny is only ever an adjective. As such it can function as attributive modifier of nouns (e.g. a tiny flower) or predicative complement of verbs (e.g., that flower is tiny), and a few other minor functions.

The situation with little is similar when it's an adjective, but it's not just an adjective. We also have the determinative little (e.g., little water) and the complex determinative a little (e.g., a little water; a complex determinative is a single determinative made up of two or more words.) Determinatives typically function as determiner of nouns (e.g., that water), but some determinatives also function as modifier of adjectives (e.g., it wasn't that big). Most adjectives, tiny included, can't function as modifier of adjectives. Also, some NPs can modify adjective (e.g., [a city block] longer).

That made me [a little] happier 

D a little is modifier of Adj happier

*That made me tiny happier

Adj tiny can't modify Adj happier

*That made me a tiny happier

D a can't form a constituent with Adj tiny or Adj happier; Adj tiny still can't modify Adj happier

a bit

D a is determiner of N bit forming an NP

That made me [a bit] happier

NP a bit is modifier of Adj happier

a little/tiny bit

D a is determiner of N bit; Adj little/tiny is modifier of N bit

That made me [a little/tiny bit] happier

NP a little/tiny bit is modifier of Adj happier

The interesting question is how little became a determinative and how a little became a complex determinative.

Quite simply "a little" is used as an adverb and a determiner, but "a tiny" is not used at all.

Both little and tiny are used as adjectives.

It seems to me that "happier" is a (comparative) adjective

and

"a little" is a compound predicative intensifier which acts on that.

"A tiny", however, is not a compound intensifier of any kind, but a determiner plus an adjective, despite the fact that "tiny" and "little" are synonymous in their simple adjectival forms.

In other words, "a" and "little", can be put together to create an intensifier, while "a" and "tiny" cannot, despite the similarity of meaning between "little" and "tiny".