There is of course a direct relationship between the engine-out divert speed and the altitude you are able to maintain. The faster you fly, the lower will be the maximum single engine altitude. Boeing applies some pretty hefty icing penalties in the performance manual. It is quite possible in the 767/757 fleets, if the TAT falls within the correct range, to have a published single engine capability of less than 10000 feet at higher divert speeds after the icing penalties are applied.
So one question worth considering is the temperature/moisture content at the diversion altitudes and the obstacle clearance requirements. A higher diversion speed results in a higher TAT but a lower altitude; the inverse may be a more desirable option in some cases.