Percolation network analysis with 1H MAS solid state NMR
Solid-state (MAS) NMR can most likely help you understand more about the interaction between the percolating additive and the resin. If you have not yet carried out solid-state NMR analysis, it can be a useful complement to information that you have determined by other methods. However, it is hard to be 100% sure that it would answer your exact question in terms of guiding you on how to modify the percolation threshold. 

Should you do $\ce{^1H}$ MAS solid-state NMR?

Unlike in solution-state NMR, $\ce{^1H}$ MAS solid-state NMR is not yet a routine procedure and has much fewer applications (compared to the widespread use of $\ce{^1H}$ in solution-state NMR) until fairly recently (within the last 5-10 years). 

$\ce{^1H}$ has a high natural abundance and high sensitivity. However, unlike in solution-state NMR where the free tumbling leads to averaging of such anisotropic interactions, in solid-state, anisotropic interactions are retained. Thus, the high abundance and sensitivity of $\ce{^1H}$ lead to a dense network of anisotropic dipolar interactions in the solid-state, meaning that a "straight-forward" $\ce{^1H}$ spectrum would yield a broad unresolved hump rather than resolved, informative (sharp) signals.

Thus, the approach taken more commonly is to detect heavy atoms with better inherent chemical shift resolution, such as $\ce{^13C}$. Techniques such as cross polarisation (CP) to tackle the problems of lower sensitivity with heavy atoms have been around for decades and can now be considered routine in solid-state NMR. So, if you may want to consider $\ce{^13C}$ CP-MAS solid-state NMR instead of $\ce{^1H}$ MAS solid-state NMR.

What access do you have to $\ce{^1H}$ MAS solid-state NMR facility and expertise?

$\ce{^1H}$ MAS solid-state NMR has undergone some dramatic technological improvements in the past few years. While it is still not routine, it is now possible to acquire (more) resolved $\ce{^1H}$ solid-state NMR spectra. However, highly specialised hardware is required.

The hardware required is a high magnetic field (700 MHz to 1 GHz) and very high MAS rates (at least 40 kHz but ideally 60-100+ kHz). Both of these types of hardware have been available commercially since about 2009. However, there are not many labs that I know of that have both of these pieces of hardware, and nearly all are national or pan-EU facilities (and unfortunately the ones I can think of nearly are nearly all focussed on structural biology). It is also not straightforward to operate these hardware, so you will need to interest these labs in a collaboration.  

There are some labs that are trying to do $\ce{^1H}$ MAS solid-state NMR with lower magnetic fields and/or MAS rates by developing specialist pulse sequences to carry out the $\ce{^1H}$ homonuclear decoupling required. It is not an area of ssNMR that I am very familiar with though I believe that there are people doing this in Israel and India.

Generally, the groups doing $\ce{^1H}$ MAS ssNMR are not going to be doing it like a "service". You cannot just give someone a sample and expect them to give you a spectrum. If you can find a group that specialises in $\ce{^1H}$ MAS ssNMR to collaborate with, then it is worth doing $\ce{^1H}$. Otherwise, if you find a ssNMR group with other interests, you may be able to ask them to carry out a (relatively routine) $\ce{^13C}$ CP-MAS spectrum for you.

Sample amounts and types

I know you did not directly ask about this, but I feel it may be helpful to give an idea of how much and what types of sample you will likely need.

For very fast MAS spinning and $\ce{^1H}$ detection, you need only a small amount of sample. Depending on the density of your polymer it is likely that no more than 10 mg is required. (My estimates are based on biopolymers - mostly carbons and protons, so adjust as appropriate for your polymer.)

For heavy atom detection, more sample is usually advised because that will help with the lower sensitivity. Around 50-100 mg would be a good amount. 

It will also be helpful to you (and your collaborator) if you can provide the monomer of the resin, and the pure additive (as monomer if necessary). It can be helpful if you have several rubbers that are more simple "models" of what you are trying to investigate (known to have one particular type of polymer links, etc). 

...and what information can you expect to obtain?

In the best case scenario, you will have a fairly resolved spectrum of your polymer sample, and by comparing the spectrum of your monomers and your polymer you can assign nearly all the signals. By looking at which signals have shifted/broadened the most, you can determine the atoms which are interacting (or have become bonded). Knowing which atoms are interacting, you can then have some guidance on what functional groups may allow you on to adjust the percolation point as you wish.

But! If you are used to looking at solution-state NMR spectra, solid-state NMR spectra can look incredibly broad. For polymers, the resolution is often poor and signals are fairly broad (even by solid-state NMR standards). In my experience (detecting $\ce{^13C}$) by CP on a 400 MHz spectrometer with a MAS rate of 10 kHz), the linewidths of polymers I have seen are around 150-300 Hz. 

The resolution is a very sample-dependent parameter, and is one of the main reasons why I cannot answer definitely whether ssNMR will help you or not. If, as you said, the additive and the polymer are very similar, their signals may well overlap or be insufficiently resolved for you to determine if there has been any changes.

Moreover, polymers are often heterogeneous. This means that one signal in the monomer may broaden and split into multiple signals in the polymer. If this happens at too many atoms, it will make it difficult to assign your polymer spectrum.

There are ways, again, to overcome these problems. It is possible to employ multidimensional ssNMR spectroscopy detecting $\ce{^1H}$ and/or heavy atoms, leading to heteronuclear correlations or homonuclear double-quantum colrreations. The additional dimension often improves resolution and therefore helps with assignment. However, sensitivity will most likely suffer. Alternatively, the molecular dynamics within the polymer can give insight on where the interaction may be occurring in cases where you have sufficient resolution to assign but it is unclear where the interaction has occurred from observing the chemical shift alone. These approaches are again going into non-routine territory where you will need to find a collaborator rather than having it as a "service".

Here is a review if you are interested in some examples of recent multidimensional $\ce{^1H}$ MAS ssNMR work on polymers. Here is a more dated paper which shows how dynamics can help in identifying the nature of interactions at a rubber-filler interface.