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Try the Kozai mechanism

The basis for the Kozai mechanism is that in a binary system, a third orbiting body - with a much lower mass than the other two - has a quantity that doesn't change in time: $$L_z=\sqrt{1-e^2}\cos i$$ where $e$ is the orbital eccentricity and $i$ is the orbital inclination - the amount it tilts in comparison to a reference plane. As the planet orbits the main body - in this case, a star - the second body perturbs it such that the planet's eccentricity and inclination oscillate in time, with a characteristic period. Both orbital elements are changing, but $L_z$ remains constant. Therefore, even though the planet's orbit is at a given point in time Keplerian, it's actually following a much more complex path in three dimensions.

There are two scenarios where this is commonly considered:

A Jupiter-like planet perturbing minor bodies, such as asteroids.
A brown dwarf perturbing the orbit of a planet orbiting a companion star.

In your case, I think adding a brown dwarf would be a good idea. If the system is largely compact with the exception of one planet, and the brown dwarf is far enough away from the star, the inner planets wouldn't be affected, and their orbits would be stable and Keplerian. The farther-out planet, with the greater period, would be far more likely to be perturbed by the brown dwarf, and would indeed execute a strange three-dimensional orbit.

Here's an example of how eccentricity changes with time, from Takeda & Rasio (2005):

^{Figure 2, Takeda & Rasio. The two curves show two different scenarios, illustrating how the Kozai timescale can be large or small.}

If you know $L_z$, you can calculate $i(t)$ from the values of $e(t)$ for any given $t$.

Example: TRAPPIST-1

The TRAPPIST-1 system is an appealing example. It is extremely compact - all seven planets have semi-major axes less than 0.07 AU. They're held stable by resonances. What if we were to put another planet in orbit at, say, 2 AU, and to add a brown dwarf companion at 8 AU?

The mass of TRAPPIST-1 is $0.089M_{\odot}$; a brown dwarf might have a mass of $0.02M_{\odot}$. Kepler's third law tells us that the planet's orbit is 9.48 years. The period of Kozai oscillations is, in the case where the brown dwarf is much more massive than the planet, $$P_{\text{Koz}}=P_p\frac{M_*}{M_b}\left(\frac{a_b}{a_p}\right)^3(1-e_p^2)^{3/2}$$ where the subscripts $p$ and $b$ refer to the planet and brown dwarf. Say the initial eccentricity is $0.5$. For us, this turns out to be $185P$, or 1785 years. In other words, noticeable inclination changes could happen in less than two millennia, which is fairly small on astronomical timescales.

The oscillations will happen quicker if

$M_*/M_b$ is small, which means a low-mass star (like a red dwarf) is a safe bet for the primary, or
$a_b/a_p$ isn't too large.

If we substituted the Sun in for TRAPPIST-1, the effects would be much different, and the period of oscillation would increase by more than an order of magnitude, producing less drastic changes.

The maximum eccentricity is a function of the initial inclination of the planet: $$e_{\text{max}}\simeq\sqrt{1-(5/3)\cos^2(i_0)}$$ If, say, $i_0=75^{\circ}$, we find that $e_{\text{max}}=0.94$, which is enormous. We then calculate that if $e_0=0.5$ and $i_0=75^{\circ}$, $L_z=0.058$. We can then graph $e(t)$ and $i(t)$: