Is possible to use this system when working with bacteria-infected cells that are susceptible to Dox, or it is better to use other system of gene expression induction?
The tet system is a good choice for that kind of project because the tetracyclines are easily going over all kinds of membranes. Most other induction systems require carrier-dependent inductors (lactose, arabinose, etc) that will make it difficullt to penetrate host and internal bacteria membranes alike. If your bacteria are too sensitive to the usually used low-toxic chlorotetracycline (=aureomycin), you may have to find another tetracycline variant that induces the promotor without harming the cells. In any regard, doxycillin sensitivity is no valid parameter to estimate the chances of success.