A single electron does not experience Coulomb repulsion with itself. The potential and wavefunctions you describe involve only one-electron, so the state energy cannot be associated (even as an approximation) with the Coulomb repulsion between two electrons.
On the other hand, the double delta potential can be seen as a very crude approximation of the Coulomb potential of two positive nuclei. The fact that for the bonding state, the electronic charge density piles up more in between merely reflects that the electron is more likely to be found between the two nuclei, justifying the name "bondig". The opposite holds for the anti-bonding state.
For the sake of argument, let us add another electron to your model and ask what is the ground state of the two-electron systems?
Without explicitly stating other assumptions, you model again does not imply any interaction between the electrons beyond Pauli exclusion principle. If these electrons were of opposite spins, they would both occupy the lower-energy bonding state. If they were of the same spin, one would be in the bonding state and the other in the anti-bonding one.
Finally, let us ask if we have two electrons of opposite spins in the bonding state and we then "turn on" the Coulomb repulsion between the two, what would happen?
It depends on whether the Coulomb energy between two bonding states is larger than the energy increase resulting from moving one or both electrons to the anti-bonding state. If it is not, the electrons stay in the bonding state and you get a covalent bond.