Step 1: Define Constraints
$p + q + r = 900 \implies p + q = 900 - r$.
$0.3q \le p \le 0.7q$.
$r \in \{169, 196, \dots, 484\}$ (Perfect squares between 150 and 500).

Step 2: Finding Maximum $p$
To maximize $p$, we minimize $r$. Let $r = 169$.
$p + q = 731 \implies q = 731 - p$.
$p \le 0.7(731 - p) \implies p \le 511.7 - 0.7p \implies 1.7p \le 511.7 \implies p \le 301.11 \dots$
The maximum integer value is $p = 301$.

Step 3: Finding Minimum $p$
To minimize $p$, we maximize $r$. Let $r = 484$.
$p + q = 416 \implies q = 416 - p$.
$p \ge 0.3(416 - p) \implies p \ge 124.8 - 0.3p \implies 1.3p \ge 124.8 \implies p \ge 96$.
The minimum integer value is $p = 96$.

Step 4: Final Calculation
Sum of Max and Min values $= 301 + 96 = \mathbf{397}$.