Offline attribution is tough. So much so that some of the best performance marketers I know won't touch offline channels - even when there are clear eamples of DR teams (Credit Karma, Smile Direct, Peloton) making them work.
There are two common approaches I've observed or heard repeated over and over again:
1) Spot Measurement - This method plots on-air time (works better for TV, radio) against conversions in a time series and measures the lift. Ie, what did conversions look like 5-10 minutes after the spot ran vs the 5-10 minutes before and what was the subsequent lift. It's not perfect, and gets tougher for multi-touch and/or high consideration funnels, but it's one of a few helpful data points.
2) HDYHAU - This method surveys customers after they purchased with a quick "How did you hear about us?" form. It then compares the % breakdown of the survey (which includes offline channels as an option) to the direct attribution (click-based) and uses the diff to assign credit.
Your question may have been rooted specifically in the scenario where multiple offline channels are running at once, which certainly adds to the complexity. Hopefully the above serves as a helpful look at how other teams may be looking at the problem.
Offline can be daunting for DR teams, but that certainly provides an advantage to the teams that can make it work and an argument for taking a well-guided shot.