It's not just minimum wages. If productive sector employees' demand for goods causes inflation, prices will rise. If you want to keep your nurses/doormen/teachers you'll have to increase their wages regardless of what the minimum wage is.
That might just mean more competition is created as there's a bigger market to chase. Minimum wage has the same risk as the one you mentioned, but also all but guarantees higher prices (particularly on labour-intensive goods such as food) as everyone's wage needs to go up to make the harder/worse jobs worth doing.
