You can take $E$ to be the union of $B$ and $C$, or actually any set that contains $B\cup C$. Now, your concerns about composition are a bit subtle and their solution depends on the "tiny" details in the definition of "relation" and "function". For my definition, both a function and a relation $f\colon A \to B$ are defined as a triple $(A,B,f)$ where $f$ is a suitable subset of the product set $A\times B$. So, it is imprecise to say that $f$ is a also a function $f\colon A\to E$ if $E$ contains $B$. What is true (or more precise) is that, in your situation, if $R_1=(A,B,R_1)$ and $R_2=(C,D,R_2)$ are two relations with $B\neq C$, for any set $E$ containing $B\cup C$, there are two uniquely induced relations (so, uniquely determined but different, in principle, from $R_1$ and $R_2$ if $B\neq E$ and/or $C\neq E$), $(A,E,R^E_1)$ and $(E,D,R^E_2)$. Then, for any such $E$, the composition $R_2^E\circ R_1^E$ is well-defined and it does not depend on the specific choice of $E$, but only on the original relations $R_1$ and $R_2$. Using this invariance, you can extend the definition of composition and define $R_2\circ R_1:=R^E_2\circ R^E_1$ for any $E\supseteq B\cup C$. Again, this will be well-defined because the composition $R^E_2\circ R^E_1$ does not depend on the choice of $E$. But still, with the standard definition of composition you cannot make sense of an expression like "$R_2\circ R_1$" if the target of $R_1$ is not contained in the source of $R_2$.
EDIT: As professor Velleman made clear in the comments, it seems that there is no complete agreement on the definition of basic concepts like a "relation" or a "function". My research in the last 10 years has been mostly in category theory, so the above definition of a function is certainly the most natural to me, and to other people with my same background. On the other hand, it seems that people with other backgrounds prefer to give slightly different definitions (a relation between $A$ and $B$ is not a triple, but it is really just a subset of $A\times B$, so the notion of source and target become not well-defined, but otherwise everything still works just fine). These two approaches are both present in the literature and in textbooks, so they are both to be considered correct. The important thing is to make sure that we adopt one of the two conventions and maintain it consistently throughout our work. This is exactly where my answer above becomes wrong: I have given an answer that uses the definition of a relation as a triple to a questions that assumes that a relation is, by definition, just a subset of the product space.
As suggested by prof. Velleman I will leave this answer anyway because it will certainly be instructive: it shows how important foundations are, and how fast a statement can become true or false due to the slightest and apparently innocuous change in a definition.