What part of speech is "telling" in "that would be telling"?

Solution 1:

In That would be telling, there is not enough information to distinguish a noun from a gerund.
Telling could be either one in this sentence. Which is, incidentally, a fixed phrase, so it's irregular.

Gerunds act like nouns, but they have verbal powers, like the power to take a direct object.
Real nouns, that have made it all the way from verbhood, have the power to take articles.
That would be telling contains neither a direct object nor an article.

Solution 2:

The participle telling "that tells [something], that expresses [something]" has turned into an adjective meaning "expressive, revealing". This can be seen in what is clearly the adjectival use:

For a telling tale of the ways in which women in England deployed campaigns for indigenous women's rights to their own ends see [book] ... — Source.

A telling case is not a representative case, but one that allows in-depth exploration of theoretical issues not previously visible. — Source.

... about your characters; and deliberately, from your conscious writing self, who already knows the character very well and must divine the most vivid way to convey that knowledge to the readers. How do you deliberately create a telling detail? — Source.

It has the same meaning in your example, so it is also the adjectival use.

Solution 3:

It is a verb in the progressive. Technically, it's part of the verb phrase: "would be telling", which has modal verb (would) followed by an auxiliary verb (be) and the progressive main verb (telling)