The p-order method got a lot of attention in the solution of Askey's 1986 problem 6514 in the Math Monthly to show that 
    $$\frac{(3m + 3n)!(3n)!(2m)!(2n)!}{(2m + 3n) !(m + 2n) !(m + n)!m!n!n!}$$ 
is always an integer. 

It had been conjectured that this is the constant term of 
    $$\left( \prod{(1-u/v)} \right)^m \left( \prod{(1-uv/w)} \right)^n$$ 
where each product is over the 6 ways to set the variables to x,y and z (and hence is an integer). This was established in:  A Proof of the $G_2 $ Case of Macdonald's Root System-Dyson Conjecture by Doron Zeilberger, SIAM J. Math. Anal. 18, 880 (1987), DOI:10.1137/0518065 . So this is certainly not a p-order proof. However I don't know that there are constant term identities for these other ratios. The article does cite (a special case of) a theorem of Morris  showing that the following expression is a constant term and hence an integer:  
    $$\frac{(a+b+2c)!(a+b+c)!(a+b)!(2c)!(3c)!}{(a+2c)!(b+2c)!(a+c)!(b+c)!a!b!c!c!} $$