On the relationship between income distribution and GDP
You assume that higher spending causes higher GDP. This is not necessarily true. 

Saving income will increase GDP through investments (unless you're in a Keynesian trap). Think about the most standard growth model, where the future (and steady state) GDP increases strictly in the savings rate.

Hence, whatever increases the savings rate (in your toy model, higher inequality), potentially increases future GDP. This was an argument brought forward in Barro (2000). Version with linear savings rate are in Bertola et al (2006) 

Of course, this is the neoclassical answer. There are many reasons why high inequality decreases growth/GDP which work through political economy models or similar. I'll list some of them despite you not asking explicitly for them, perhaps it's of use for future visitors:


Fiscal Policies: Equality leads to less necessary government redistribution and hence more incentives for investments (Meltzer and Richard, 1981; Corcoran and Evans, 2010; Persson and Tabellini, 1994)
Crime: Inequality reduces opportunity costs for illegal activities that harm GDP (Alesina and Perotti, 1993; Barro 2000)
Imperfect Credit Markets: Fixed investment costs or similar may lead to higher growth in unequal societies (Barro 2000)
Saving Incentives: With "not much to lose", poor people face moral hazard in unequal societies, a mechanism that decreases growth with inequality (Banerjee and Newman, 1991)
Fertility-Education: Equality in human capital leads to smaller fertility rates, and hence lower GDP. The Argument is too long to be summed up here; see Perotti (1996) and Croix and Doepke (2003))