CUs operate differently from banks a lot of them want POI when you join and app for credit. Paystubs are what they normally request.
Good luck with your Alliant application! My Visa Platinum card approval was instant in July, so I can't speak to the specific message you received, but it would help to know how much you requested for SL.
The Alliant application requires you to list a specific amount for SL and they are very conservative in general with SL. If the amount you requested is higher than they feel comfortable granting, in my experience they turn it down outright rather than countering. (I applied when I first became a member early in 2016 and was denied then; re-applied in July and requested a lower CL for the card and was approved instantly.) Your mileage may vary, of course.
Alliant also uses a Pinnacle scoring system which is different than any other lender I've experienced. Pinnacle scores can be as much as 20-50 points lower than FICO scores.
Overall, I like Alliant very much and hope it goes well for you. Their customer service is outstanding and their checking and savings products are great. Also, they gave me 0% for purchases for 18 months on my Visa Platinum, which I appreciate but is negated a bit by a low SL on the card itself.
I also have an unsecured loan with Alliant and I have prepaid it through May - they have a wonderful feature where they allow you to pay months in advance and I won't have another payment due until May! I took out the unsecured loan primarily to develop the relationship with them further, and also to maximize the HP from the credit card.
Alliant was able to use the same HP for both the credit card and the unsecured personal loan. If you wish to combine two products into one HP, you will need to call in for the second product within 30 days and specifically request them to use the original HP.
They are a very conservative credit union so if you really want a product with them, nurture the relationship and don't give up. I hope it all works out! 