A vertical variation of modern versions of Michelson-Morley

Firstly, you need to calculate how much your hypothesised effect will change the optical delay in each of the interferometer's arms and check that you expect to see any result with your proposed experiment. Otherwise put: what are the specifications of the interferometer (arm lengths, light source requirements etc, vibrational tolerances) that will let you see your effect if it is real and are they reasonable?

I can see a major problem with your setup which you will need to overcome. There are several effects which I can think of which will influence your experiment. The first is the change in optical length of a given interferometer arm that arises through the gravitational redshift of the light propagating between different gravitational potentials when the arm is vertical (as verified by the Pound-Rebka experiment - see wiki page of this name as opposed to the absence of this effect when the arm is horizontal. This effect is small, but it can be precisely calculated from the Schwarzschild Metric and it is repeatable. So this effect is not a problem for you. A second effect is the gravitational tidal effect which is owing to the second order variation of the Schwarzschild Metric). Indeed, this is an extremely mild version of spaghettification. The interferometer responds to this by assuming a strained state - it is equivalent (in the Newtonian limit) to the strain required to counteract a variation in gravitational acceleration given by $3\,g\,\Delta/R$ for a vertical distance of $\Delta$, where $R$ is the Earth radius when the interferometer is in freefall. That is, a few micro-g per meter of vertical distance. Again, this is a repeatable effect.

An effect which is going to be vastly bigger and IMO almost impossible to account for is the change in an optical arm's length through the weight-induced mechanical stress on the arm as the arm of the interfermeter is rotated from horizontal to vertical. And the interferometer must be rotated as in the Michelson-Morley experiment as there is no non-interferometric way to compare the optical lengths of the interferometer arms. The solution is to swap the arms' roles (by rotation) and check for the effect thus.

So this really means your experiment needs to be done under conditions of weightlessness. Does your proposed effect still exist in freefall according to your theory? Otherwise, you will need to develop a repeatable, independent-of-light means of measuring the weight induced strain in your interferometer arms to hundredths of wavelength accuracy.

Any custom built interferometer system of the precision you will be seeking will easily eat up tens of thousands of US dollars / euros at the time of writing (2014) in the optics production and the mechanical alignment system. You will need to become highly adept at mechanical design and production of engineering drawings to get what you need. Add \$10K for either the acquisition of software to help you do this (e.g. Solidworks) or professional engineering help (in the latter case, add \$20K to \$30K).

Now, add the cost to put your experiment into low Earth orbit. The Space Exploration Stack Exchange Question "What is the current cost-per-pound to send something into LEO?" may help you. It would be a fair bet, from the figures quoted there, that your looking at \$25K USD / kg. So a ten kilogram interferometer satellite system (don't forget data telemetry) is going to cost you a quarter of a million USD / euros to get it to where it will work.

So all up, I'd say a budget of the order of \$400K USD / \$400K euros is looking like a minimum figure.

An example of a similar experiment is the famous measurement of a gravitationally induced phase shift in a neutron beam by Colella, Overhauser, and Werner (often called "the COW experiment"). It's interesting to note that while there was an unambiguous gravitational phase shift, its size was not as predicted.