Depending on what you actually need for your application, there might be something useful already known. Do you really need a tricategory, as opposed to some other model of weak 3-categories? Even if you had that, would you be able to do much with it?
There are several equivalent ways to think of profunctors, and some of them lend themselves to categorification quite easily. One is this:
A profunctor from $C$ to $D$ is the same a a colimit-preserving ordinary functor between the presheaf categories $PSh(C)$ and $PSh(D)$.
(See here for details: http://ncatlab.org/nlab/show/profunctor#FuncsOnPresheaves)
That's good, because a lot is known about categorifications of categories of presheaves and of morphisms between them.
For instance it is straightforward to set this up over bicategories: take objects bicategories, and hom-bicategories to be the full sub-bicategory on bicolimit-preserving bifunctors between their bipresheaf categories.
In case that your application is such that you only need higher categories whose higher morphisms are all invertible, one can go much further and consider the (oo,1)-category of (oo,1)-profunctors. By the above, this is simply the gadget whose objects are small (oo,1)-categories and whose hom-oo-groupoids are the full sub-oo-groupoids on the (oo,1)-colimit preserving (oo,1)-functors between the corresponding (oo,1)-presheaf categories.
More generally, one can generalize here (oo,1)-presheaf categories and all in addition all their reflective sub-(oo,1)-categories. That structure of (oo,1)-profunctors has proven to play a major role as kind of categorification of the category of vector spaces: one thinks of a presentable (oo,1)-category as vector space, of colimits as being sums of vectors, as colimit preserving (oo,1)-functors as linear maps.
More on this is here: http://ncatlab.org/nlab/show/Pr(infinity,1)Cat .
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