Is bounded waiting satisfied in the 2 Process Solution?
A little remark first: The bounded waiting (BW) property is defined with respect to algorithms, which is in turn defined as a set of concrete executions. Thus, we cannot conclude whether an algorithm satisfies BW if a specific execution does.



For your question: For the specific execution in which only $P_1$ takes steps, BW is satisfied. However, it is not because "there exists a bound on the number of processes that may enter after process desires to enter CS". Instead, BW is satisfied simply because $P_2$ does not desire to enter the critical section. See the following definition from Mutual Exclusion (wiki).


  A $k$-bounded waiting property can provide a bounded waiting guarantee and it requires in every execution, a process can only enter the critical section at most $k$ times while another process is waiting to enter the critical section.


A more precise definition can be found in the Book: The Art of Multiprocessor Programming; Exercise 9 in Section 2.10:


  Define $k$-bounded waiting for a given mutual exclusion algorithm to
  mean that if $D_A^j \to D_B^l$ then $CS_A^{j} \to CS_B^{l+k}$.


Here, $D_A^j$ means process $A$ finishes its doorway section for the $j$-th time, and $CS_A^j$ denotes the interval during which $A$ executes the critical section for the $j$-th time.

In the execution above, $D_{P_2}^j \to D_{P_1}^l$ does not hold because $P_2$ does not take any steps at all. Thus, BW is trivially satisfied.

Suppose that $P_2$ has finished its doorway section (i.e., $D_{P_2}^1$ holds) and then halts. If $P_1$ continues entering and exiting the critical section, then BW is broken.



A little remark again:


  I think BW is satisfied, as there exists a bound on the number of processes that may enter after process desires to enter CS.


It seems that you have a misunderstanding about $k$-bounded waiting property. $K$ here is about the number of entering (and exiting) the critical section, instead of the number of processes.