I am not denying that the standard expositions are quite terse and strongly contribute to the confusions, but your question is also full of self-inflicted confusions.

According to the de Broglie relationship, the wavelength of an object is inversely proportional to its mass. As a result, protons possess a smaller wavelength than their constituent quarks and gluons.

de Broglie wavelength

This is just straight up wrong. The de Broglie wavelength, when not modified by any qualifier, always refers to $$\tag{Definition of de Broglie wavelength}\lambda_\text{dB}=\frac h{|\vec p|}$$ And this has to be distinguished from the thermal de Broglie wavelength $\lambda_\text{th}$.

This means that when the composite particle is not moving, its $\lambda_\text{dB}\to\infty$ while its constituents are moving and so their de Broglie wavelengths would be smaller, and thus your assertion is just wrong.

Compton wavelength

The relationship that you are talking about is actually the Compton wavelength $$\tag{Definition of Compton wavelength}\lambda_C=\frac h{mc}$$ I can fully understand that the standard treatments give this quantity a mysterious flair. Its interpretation is usually given as on the Wikipedia page about the limitation on measurement and its geometric interpretation.

There is nothing wrong about those standard interpretations. They are undoubtedly correct. But there is a lot more to say and make sense of them.

If you study closely how the Compton wavelength goes into the physics, how it manifests itself in quantum phenomena, then you will slowly realise that, really, this is not a wavelength at all, but rather it is an oscillation period. The oscillation period of a quantum particle at rest, composite or not, is given by the Compton wavelength divided by the speed of light in vacuum. This represents a lower bound on the oscillation frequency of the quantum particle, the energy-time counterpart to the usual de Broglie wavelength that is about momentum-position relationship.

And because it is a period and not a wavelength, it just so happens to not be what you think it is doing. It is perfectly ok for the oscillation frequency of a composite particle to be greater than that of its constituent parts.

If it's due to the forces, does the concept of a "wavelength of a quark" still apply, or is it only relevant for protons?

It has been pointed out to you multiple times that forces are the first to die in the quantum revolution and there is no future scenario in which it makes a comeback. Stop using the force concept. It has no place in this discussion.

I have already talked about the de Broglie wavelength of the constituents in the answer above.

and If quark wavelengths exist, can we model a proton as three independent quarks influenced by forces that mimic their interactions, allowing for a linear summation of patterns?

The successful modelling of a proton had to have many many quarks and gluons, because the fundamental strong interaction is, as its name suggests, strong. It is just that there are 3 more quarks than there are anti-quarks, and with a lot of cancellation, you are left with those 3 quarks.

There is also another model of a proton as 3 composite quarks thingy that are moving slowly and exchange identities using pions. That is good for low energy low-order perturbative estimations. Not too horrible.

The quark wavelengths have very little to do with either model, though, of course, in the model that goes directly into the fundamental strong interaction, it is programmed into the code to be properly taken into account.

linear summation of patterns

make no sense whatsoever.