Continuously feeding an evaporating micro-black hole?

Yes, it would remain stable.

But these black holes are incredibly hot and there is no known matter that could feed them fast enough to balance the mass loss that they experience through evaporation. Assuming that with 'micro black hole' you mean a TeV-sized black hole, the mass loss is of the order of a TeV/fm. In addition these black holes are also small, which is to say their cross-section is extremely tiny (of the order fm^2). Not even nuclear matter (ie matter that you would find in a Neutron star) would feed them quickly enough, not even when it's relativistically compressed by a very high relative velocity.

Micro black holes have been hypothesized in some large dimension string phenomenological models and are searched for in the experiments at the CERN LHC.

The first approach to the decays was thermodynamic with Hawking radiation diminishing them rapidly. Their lifetimes are very short so there is no way to gather and contain them and experiment with feeding them in order to survive. They are supposed to occur within the frame of the 10^-23 seconds of the strong interaction, or order 10^-25 seconds.

The article linked to proposes the possibility of quasi stable mini black holes:

It has been postulated that black holes could be created in particle collisions within the range of the available energies for nowadays colliders (LHC). In this paper we analyze the evaporation of a type of black holes that are candidates for this specific behaviour, namely, small black holes on a brane in a world with large extra-dimensions. We examine their evolution under the assumption that energy conservation is satisfied during the process and compare it with the standard evaporation approach. We claim that, rather than undergoing a quick total evaporation, black holes become quasi-stable. We comment on the (absence of) implications for safety of this result. We also discuss how the presence of black holes together with the correctness of the energy conservation approach might be experimentally verified.

Even in this model scenario if one could get hold of the kernel of the mini black hole at the LHC I can not think of a way to keep it as a target in order to feed it extra energy. It will be a very soft missing particle that could not accrete matter by itself due to the size of its horizon ( as the link describes):

On the other hand, the black hole could accrete matter by colliding with atomic or sub-atomic particles in its way through the Earth. However, it has been shown [25] that accretion through both mechanisms would be only appreciable for black holes with horizons much bigger than the ones considered here (whose horizons in their quasi-stable phase satisfy 0< R0.10^−19m≪L). Therefore,we arrive to the expected result that the long lived black holes predicted by the energy conservation approach can not be dangerous.

The most I can think experimentally is if these mini black holes give signs of existence, through their first stage of thermodynamic decay and unexplainable missing mass, to set up a system to detect them at some distance from the target . Emulsions maybe: it will work if they are charged, I do not know about being neutral (hair tracks maybe from accretion of electrons). It would be a new field of experimental research. Then if one finds charged ones are being created, set up a system to trap them and use them for scattering experiments which would show their rate of accretion too. From the model in the paper it seems to be small, and feeding them to keep them from totally disappearing seems to me to be impossible, but a calculation would be needed to support this statement.