A pretty fundamental question:
why the Gresham Dollar and not regular dollars?
I mean, there's no practical limitation on the monetary authority creating more real dollars to pump into the economy. The reason it doesn't (and the reason why it's fashionable to separate monetary authority from fiscal authority) is because of beliefs about pervasive inflation caused by large increases the supply of money, particularly if that comes in the form of handouts to those with a high marginal propensity to spend it.
But as far as I can see based on your outline, Gresham dollars are at best functionally equivalent to regular dollars in their effects on demand in the economy. Demand increases when they're handed out, supply lags, ergo prices go up.
But you can add in a [very] slight increase in cost of sale for merchants accepting GD, especially if they need to convert to regular dollars to pay their suppliers.
So why Gresham dollars? Why not just hand out newly minted regular dollars? Or food stamps, if the aim is to socially engineer how they're spent?
As far as I can see, all the differences are disadvantages (confusion, costly new infrastructure, discouraging saving behaviour amongst low income beneficiaries)