That's good news and bad news, you are right.
Good because you know where the issue is, bad because it's a custom theme that we cannot help with debug.
But let me not abandon you here.
I'd as first make a copy of the theme (just copy the entire theme to a new folder).
Then dissect it. This means to target particularly JavaScript codes and CSS codes. I'd as first open that theme's functions.php and look out for wp_enqueue_scripts() code.
Comment those you find (comment in PHP means to put a // before the code, so it is grey in your editor).
That will effectively remove those scripts from the theme, and each time you do so, test again.
Of course, you shouldn't remove the one that enqueues your SlickNav codes as that is what you want to test.
I think this should quickly show what's conflicting with your scripts.
Another approach can be to "rebuild" the Custom theme.
The approach is similar but instead of copy the existing theme to a new one, you'd start a fresh theme and start only with index.php and functions.php.
Bit by Bit insert code from the old theme to the new one.
Start as soon as possible with inserting the SlickNav and the very minimal code you need for that to work (which will be a WordPress menu support and some basic init code for the theme)
Again check when the issue comes back.
Please also always keep your browser console open (to show eventual JS errors) and activate WP Debug to show every error eventually thrown (https://codex.wordpress.org/WP_DEBUG).
If everything fails, or if you narrow down the code causing the issue and are lost in how to fix it, please feel free to feedback here, because while I cannot debug your custom theme I can always put in ideas, indications and share documentations or other resources to debug it further and fix it.
Last but not least, I'd like to share a theme that you might be finding useful:
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This is a extremely minimal theme, that I built in past to work specifically with Toolset and layouts in special.
You can not only use it and adapt as you like - it's free and open source - but you can also consult it and eventually re-use knowledge gained from for your own themes.
Please acknowledge that this is not an OnTheGoSystems product and that just as for other custom themes we cannot support it here in the Forum. But, in all honesty, the theme is so minimal I had to make 2 adjustments since it was created - and those weren't bugs, just adjustments. This means, it's a very stable theme that has almost no conflict potential.
The downside to this theme is it uses Bootstrap 3, and isn't ready for Bootstrap 4 yet (if I find time in future I'll make adjustments in regard).
I do hope this helps and please do not hesitate to ask for any information or help - the worst answer you can get is "we can't help", but we always try to step a bit over that line and at least give recommendations or ideas.
Thank you for continuing with us and Toolset 🙂