Thanks for replying. (By enthropy I assume you mean entropy, not enthalpy.) The entropy of the surroundings seems to rise by different amounts in the two different situations, but the heat emitted by burning a given amount of fuel will be the same, i.e. ΔH. The problem is getting the equality to work in different situations.
Suppose when no energy is stored, the surroundings are at 25 deg.C and the heat flow, ΔH, increases the entropy of the surroundings by ΔS. In the second case suppose 20% of the fuel's energy is stored in a battery using an alternator.
If the engine (i.e. the system) is kept at the same temperature in both cases and the Gibbs energy stays the same then TΔS is constant (from G = ΔH - TΔS.) But only 80% of the heat now contributes to the entropy of the surroundings. So to keep TΔS the same, the surroundings need to be 25% hotter. By my reckoning this means the temperature needs to rise to about 100 deg.C.
Assuming this doesn't happen, and also assuming that the Gibbs equation gives the right answers, it seems obvious that it's the conversion of chemical energy into enthalpy, ΔH, which is important in determining the viability of reactions as far as the surroundings are concerned. This is the quantity used in the equation. The change of entropy per se is irrelevant.