Ideal Gas Equation Contradiction
Your arguments do not contradict each other. You are trying to separate a process into two processes, which are coupled of course, but in first approximation this is a good way at looking it.

Let's state our model system again: A balloon with the pressure $p_1$, the volume $V_1$ is heated from temperature $T_1$. As a result we see an inflated balloon. Let's call those variables with $2$ as an index. Of course we treat everything as ideal as possible.

Now we look at the coupled process as two separate processes:


Isochoric heating, i.e. $V=\mathrm{const}$:
\begin{align}
&amp;&amp;p_1V&amp;=n\mathcal{R}T_1\\ 
&amp;&amp;p_2V&amp;=n\mathcal{R}T_2\\ 
\therefore 
&amp;&amp;\frac{p_1}{p_2}&amp;=\frac{T_1}{T_2}\\ 
\therefore
&amp;&amp;p&amp;\propto T\\ 
\end{align}
That is Gay-Lussac's law.
Isothermal expansion, ie. $T=\mathrm{const}$
\begin{align}
&amp;&amp;p_1V_1&amp;=n\mathcal{R}T\\ 
&amp;&amp;p_2V_2&amp;=n\mathcal{R}T\\ 
\therefore 
&amp;&amp;\frac{p_1}{p_2}&amp;=\frac{V_1}{V_2}\\ 
\therefore
&amp;&amp;p&amp;\propto \frac{1}{V}\\ 
\end{align}
That is Boyle's law.


You can also look at the whole process as the pressure stays the same while the balloon is heated. That then is an isobaric process, i.e. $p=\mathrm{const}$:
\begin{align}
&amp;&amp;pV_1&amp;=n\mathcal{R}T_1\\ 
&amp;&amp;pV_2&amp;=n\mathcal{R}T_2\\ 
\therefore 
&amp;&amp;\frac{V_1}{V_2}&amp;=\frac{T_1}{T_2}\\ 
\therefore
&amp;&amp;V&amp;\propto T\\ 
\end{align}
That is known as Charles' law

Of course in reality there will be a mixture happening as the rubber of the balloon will expand, but the pressure that is possible is dependent on the material of it. In combination you will arrive at:
\begin{align}
&amp;&amp;p_1V_1&amp;=n\mathcal{R}T_1\\ 
&amp;&amp;p_2V_2&amp;=n\mathcal{R}T_2\\ 
\therefore 
&amp;&amp;\frac{p_1V_1}{T_1}&amp;=\frac{p_2V_2}{T_2}\\ 
\therefore
&amp;&amp;\frac{pV}{T}&amp;\propto \mathrm{const}\\ 
\end{align}