Framer,
LRC on the B744 corresponds to CI 250. I expect the OP would have been using a cost index somewhat less than that, as most operators are pretty fuel conscious these days. So, yes it would be an acceleration from ECON to LRC.
OP,
When you said you were too heavy for F350, I assume that your FMC max altitude was less than that and your flight plan was based on the correct weight. You may be able to make a cruise CG correction to increase your max altitude because it usually defaults to a conservative value. When pushing close to max altitude, be aware of the buffet boundary/expected turbulence, and OAT trend.
Other than that, you are best off to just wait until you burn down to an appropriate weight. In fact, unless there is a strongly increasing tailwind you should be close to optimum altitude anyway.
Basically the idea proposed is just fiddling with cost index, which doesn't make any sense because the fuel/time costs remain the same for the entire flight.