Can't help with Bordieu, but perhaps with Kant. Requisite, a key concept:
Where does a circle exist? In a surface. So,
- Ontological perspective: part of the conditions of possibility of a circle is the existence of a surface.
- Epistemological perspective: part of the conditions of possibility of the knowledge of a circle is the knowledge of a surface.
In the Kantian terminology, the adjective transcendental is used to refer not to the object, but to the (epistemological) conditions of possibility of the object (memorize that! such dialectic pair is essential to understand the term).
When Kant writes about the transcendental idealism, it is not talking about idealism, but about the conditions allowing the possibility of such idealism.
Notice that Kant refers to the transcendental from an epistemologic standpoint (that is, how is it possible the knowledge that grounds such idealism?). But in order to understand the term in a simpler way, I use a different approach in my writings: consider it from an ontological perspective (Kant rejects the ontological perspective, but it's ok for us beginners). In such case, transcendental becomes relative to the context where it exists. So, when we talk about transcendental idealism, we're not talking about idealism, but about the context where such idealism exists (so, phenomena, noumenon, etc.). See? Way more simple.
Now, what is the transcendental subject? It is the subject, in regard of the context allowing the possibility of its existence. So, me, within all the metaphysical conditions that allow me to exist as a rational object. From the Kantian epistemological standpoint, the subject, from the standpoint of the conditions that allow the knowledge of the subject.
Now, knowing such subject, consider the "universal experience of a transcendental subject", which seems trivial (I can be wrong, again, Bordieu is unknown to me).