I have an object that takes a shape similar to a catenoid, which is known for its zero mean curvature.
According to Laplace's law, the pressure difference is proportional to the mean curvature. This implies that if the mean curvature is zero, then the pressure difference is also zero.
However, I am aware that when there is a non-zero pressure difference between the inside and outside of the object, which has left me puzzled.
Is it possible to have a shape resembling a catenoid but with a constant pressure difference? Furthermore, how can I relate the curvature to the distance, r(z), from the z-axis to any point z on the boundary of this shape?
Yes it is possible. Since you are already familiar with $\Delta p=0$ catenary case I shall mention the following extension hints which I trust you can take to the next step.
From equilibrium of static forces which is also the Laplace equation for a soap bubble film / minimal surface. $N$ surface tension property, $H$ mean curvature, we have
$$ \frac{N}{R_1}+ \frac{N}{R_2}= \Delta p $$ $$ \kappa_1+ \kappa_2= \frac{\Delta p}{N} =2 H $$
A computation and plot yield progressive loops, DeLaunay Unduloids.. depending on the amount of pressure used to blow or use suction in a cut tube smeared with stearate soapy liquid as shown in various expanding or contracting bubbles of variable $H =\dfrac{\Delta p}{2N}$ yielding $CMC$ ( constant mean curvature ) shapes..
To the catenary we add a family of curves, a cylinder, sphere, corrugations, unduloids etc.
Integral of the above ode is
$$ \cos \phi = H r -\frac{C}{r} \tag 1 $$
where $C$ is an arbitrary constant. When $\phi=\pi/2 $ successive films have a vertical tangent at
$$ r_{vertical ~tgt}= \sqrt{ C/H}, R_2 \to \infty. $$ The soap bubble meridian can be obtained by integration of 1)
Special cases:
$$ H=0~ ,\text{Catenary}~;$$ $$ C=0, \text{Sphere} ~;$$ $$ ~2r_{cyl} H =1+\sqrt{1+4CH}~, \text{Cylinder} ;$$
Only a segment of noncompact CMC surfaces can be physically seen between cut tube ends during experimentation if undertaken.
Sorry for the clumsy sketch. Hope you get the idea.
Please consult also if possible the nice book by Cyril Isenberg, entitled Science of Soap Bubbles and Soap Films, a book I found fascinating on this topic.