Word, phrase or idiom for an action taken for solely political advantage

You could consider using "political gimmick". "Gimmick" means:

A trick or device intended to attract attention, publicity, or trade: 'it is not so much a programme to improve services as a gimmick to gain votes'

[Oxford Online Dictionary]

What evidence is there that Lyndon Johnson's War on Poverty was more than a "political gimmick"?

In this context, political maneuver should work:

maneuver: a clever or dishonest action that you do to get something that you want
Mercer won the election thanks to the maneuver of his son-in-law.

(Macmillan English Dictionary)

In your case you could say:

What evidence is there that Lyndon Johnson's War on Poverty was more than a political maneuver?

Another possibility is "political ploy":

Hillary Clinton Ad Calls House Inquiry a Political Ploy

(www.nytimes.com)

Consider political opportunism.

Opportunism (MW):

the art, policy, or practice of taking advantage of opportunities or circumstances often with little regard for principles or consequences

Cf. Wikipedia. In your example:

What evidence is there that Lyndon Johnson's War on Poverty was more than mere political opportunism?

I' use politics:

• (used with a sing. or pl. verb) Intrigue or maneuvering within a political unit or a group in order to gain control or power: Partisan politics is often an obstruction to good government. Office politics are often debilitating and counterproductive.'

• What evidence is there that Lyndon Johnson's War on Poverty was more than partisan politics?

(AHD)