First, as Rune pointed out in the comments, his [paper][1] with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1)$-categories (whatever those are,  ) means that the machinery is arguably very complicated. In particular, it probably doesn't satisfy your (4). It is not "effortless". 

However from what you have said I think you should have a serious look as Simpon's model of Segal n-categories, which was also mentioned by Zhen Lin. This is part of a general approach/construction for defining weakly enriched categories, and from what you have said I think there is a very good chance it applies in your case.

I won't give the formal definition. You can look that up in Simpson's book or elsewhere, but the basic flavor is that a weakly enriched $\mathcal{V}$-category $C$ will consist of the following: 

 - A set of object $x,y,z, \dots$
 - $C(x,y)$ objects of $\mathcal{V}$  for each pair of objects,  
 - $C(x,y,z)$ objects of $\mathcal{V}$ for each triple of objects.
 - $C(w,x,y,z)$ objects of $\mathcal{V}$ for each quadruple of objects.
 - etc.

Plus there are maps which are like the maps of a simplicial set. So for example there are degeneracy/unit maps like $1 \to C(x,x)$ and $C(x,y) \to C(x,x,y)$ which add identities, and there are face/restriction maps like $C(w,x,y,z) \to C(x,y,z)$ or $C(w,x,y,z) \to C(w,y,z)$.

Some derivative maps from these are required to be equivalences (the "Segal maps"). Specifically the maps
$$ C(w,x,y,z) \to C(w,x) \times C(x,y) \times C(y,z) $$
and its cousins. 

Let me describe the conceptual philosophy in the case that we are building the theory of $(\infty,1)$-categories from $\mathcal{V} = spaces$. 
Conceptually the $C(x,y)$ are the hom spaces, which parametrize the space of arrows from $x$ to $y$. The $C(x,y,z)$ are spaces which parametrize: (1) a pair of composable arrows, (2) the possible composites of those arrows (3) data witnessing the possible composite as a composite. The higher spaces are similar. 

As part of the structure there are maps

$$ C(x,y) \times C(y,z) \leftarrow C(x,y,z) \to C(x,z) $$

The left-hand arrow is required to be a homotopy equivalence of spaces (this is called the Segal condition and there are similar higher conditions). By choosing homotopy inverses to this first map (and data witnessing that it is a homotopy inverse) we get a contractible space of compositions. The Segal category builds in the higher coherence data automatically, without specifying it explicitly. 

Now the general case of $\mathcal{V}$ is similar. You need enough structure to mimic this definition. Simpson uses Cartesian model categories, but you can get away with a lot less. A version based on relative categories with products appears in section 5 of [this paper][2]. This is pretty minimal. $\mathcal{V}$ must be a category with finite products and a compatible notion of weak equivalence (plus a little bit more extra structure).  

In practice if you have a contractible space of compositions already, you can probably construct a Segal n-category. You will simply build that contractible space into the object $C(x,y,z)$. You will have to confront the higher spaces though. Hopefully your construction is natural enough that this isn't an issue. 

By the way, when you plug in $\mathcal{V}= Set$, then you essentially get the Simpson/Tamsamani version of weak n-category. The comparison of this with bicategories is describe several places, for example in Leinster's survey. Let's spell it out a bit. We begin with 0-categories which are just sets. An equivalence of sets is a bijection. This means that the Segal maps become isomorphisms. It is a standard exercise that this reproduces the usual notion of category. 

Next, on the second iteration, a weak 2-category has a set of object and then for each pair a hom category. There are also categories $C(x,y,z)$, etc. and $C(x,y,z) \to C(x,y) \times C(y,z)$ is an equivalence. To get a composition functor we must make choices of inverse equivalences to these. By looking at the diagram we get for quadruples of objects, we get an associator, and by looking at the diagram for quintuples, we see that the associator satisfies the pentagon equation.

So you can get a bicategory, but the correspondence is not quite 1-1. This is discussed in more detail in [this paper][3] by Lack and Paoli. 

As you can see from this mental exercise, for the $(n,n)$-categories there isn't really a space floating around, per se. Of course secretly there is a space, but it is hidden. This notion is much more like the classical notion of category and bicategory. Hope this helps!

 




 


  [1]: http://arxiv.org/abs/1312.3178
  [2]: http://arxiv.org/abs/1308.3574
  [3]: http://arxiv.org/abs/math/0607271