Assume a market-making model exists for original dealers that can go long and short and that this model is unique and optimal measured over each mutually exclusive bond.
Suppose, now, a new dealer may go long only, but has a long position sufficient in a bond that he can effectively deal long and short positions.
This new dealer is isomorphically equivalent to the original dealers and therefore, by contradiction, any other strategy that he employs, not the same as theirs, cannot be optimal for the specific bond(s).
Suppose, on the other hand, that the new dealer does not have a position large enough in any bond to be equivalent to the original dealers then his strategy must be adjusted and must be sub-optimal to the original dealers (since the original dealers have the flexibility to also perform this strategy but do not since their own are assumed to be optimal).
I would postulate at this point that a strategy to build long positions to create equivalence with original dealers would form a part of the new dealers strategy, but this would be counteracted most likely with a VaR restriction, i.e. the long positions would be as small as possible to permit two-way quoting.