What constitutes a disruption? Does it have to exert a secondary tidal effect rivaling Luna, or can any alteration to the current tidal pattern, no matter how marginal, be considered a disruption? Also, how measurable does this effect have to be? Thirdly, does this object have to follow the same trajectory as Luna's natural orbit? Luna's orbit around the Earth is elliptical and not equidistant. At what distance exactly are you referring to?

I'm going to assume you're taking about Newtonian physics here and not general relativity. Taking the literal understanding of your question, any introduction of any kind of mass into Earth's system will disrupt the Earth's tides and also Luna's orbit. This effect could be minuscule to the point where it cannot be measured by our instruments but the introduction of something new in the system will affect the existing system. Earth and Luna's tidal forces are also not constant as celestial objects such as the Sun or other planets also exert forces on each other at non-constant rates. The orbital path of the object and its velocity are also factors of consideration but the exact disruption will have to take place at an exact time and place.

The force of attraction (gravity) is stronger the closer you are to an object so the closer this object is, the stronger it's effect (inverse square law). To calculate the exact disruption, you will need to input the mass and distance but any mass at any distance will alter the system, no matter how small (even if it's not measurable).

We can calculate the approximate disruption of an inputted mass at an inputted distance but to be honest, it's difficult to answer your question in a meaningful way. We can try to calculate the tidal force of an object (to simplify the calculation, we input a new mass, distance, and orbital path into the local Earth-Luna system) and run the calculation from there but basically, any change will affect the system and while we can calculate the approximate change to the system as a result, in the end, we need more specific conditions (ex. assuming the introduction of a new tidally locked mass opposite of Luna's normal trajectory, at a distance of 360,000 km, how much mass is required to alter the tidal force of Luna by +/-1%?). There are also other factors to consider here such as measurement errors or rounding errors so I can't tell you the exact minimum amount of mass needed for the change but them monkeys at NASA are getting REALLY good at running these kinds of calculations (orbital mechanics).