I see there's a consistent philosophy opposing both - i.e. a fervent belief in the absolute power of markets. But what's the philosophy that says employers should be able to pay a wage that is unsustainable, while the taxpayers pick up the slack? Perhaps pragmatic conservatives recognize that a large percentage of adults making an unlivable wage is, by definition, unsustainable and might eventually result in starving peasants with pitchforks and torches (or AR-15s and cell phones). There are two options to prevent the rise of the unfed mob: market-based government intrusion, or redistributive government intrusion.
If you believe in the power of markets, then an unlivable wage is a market distortion. Why shouldn't McDonalds' prices reflect the true costs of a sustainable burger and fries? The costs and poverty impacts of an unlivable wage should be priced into products to prevent these problems in the first place. Every job should have the dignity of a livable wage. It seems pretty simple. If people don't receive a livable wage they need either government handouts or will turn to crime (19 years for a loaf of bread, anyone?). These are costs society will pay, one way or another. A market-based government intrusion (livable minimum wage) forces the market to correctly cover, rather than socializing, costs.
Granted, if all fast food restaurants raised prices their business would drop, but so what? People would decide to spend money elsewhere (or perhaps save it) rather than spend on a state-subsidized burger and fries. Realistic pricing of fast-food and retail service isn't likely to collapse our economic system (and the jobs aren't going to China). Maybe we'd see some healthier life-style choices if burgers, fries and Walmart's cheap plastic crap weren't being subsidized.
On the other hand, if you believe that the purpose of government is to help our Galtian overlords at minimal costs to them, then the EITC is your favorite solution. The EITC is inherently a redistributive government intrusion that is used to reduce the pressure for a sustainable wage. The EITC allows particular services and products to be provided at less than sustainable wages, and shifts costs to the broader set of taxpayers. The irony is that the progenitor of this idea in the US was Milton Friedman (Free to Choose your corporate subsidies?). The EITC is at its heart redistribution - but not redistribution for the good of society, but for the benefit to the few at the top. Keep the peasants just a bit above starving while spread the costs as wide as possible.
Why is the EITC popular on the right when it is inherently redistributive? It's all about government debt and tax cuts. We are subsidizing burgers and fries with debt, not by taxing today's plutocrats. The free market philosophy goes out the window if you can get the government to cover some of today's costs with government debt paid (most likely by someone else) in the future.
I suppose the EITC is better than nothing - but it's simply crumbs brushed off the table of the 1%. Anyone who really believes in the dignity of work should support the dignity of a livable wage. There's no dignity in working at less than starvation wages and getting a government handout to survive.
The EITC isn't compassionate conservatism - it's simply cheaper than a livable wage.