Short answer:
Pi-hole isn't concerned with NAT and UPnP at all.
If Pi-hole interferes with some of your UPnP HTTP requests, then that's because its doing its job: It is blocking a domain.
How do I determine what domain an ad is coming from? may help you in finding out what domain is being blocked.
Once identified, make an informed decision on whether you want to keep it blocked or not. Allowing access may just please whatever tool you use to display the text Open NAT.
Longer explanation (click)
Terms like strict NAT, moderate NAT or open NAT were coined by game console and router manufacturers, with possible contradictory or overlapping meanings (e.g. ASUS uses Open NAT as a name for a feature in its gaming routers to bundle port forwarding rules into profiles for specific games, while Xbox means its connected to the Internet the way it wants)
In networking, NAT means Network Address Translation.
While those terms are obviously related, the former usage is too ambiguous.
Pi-hole is neither a game console nor a router.
It has absolutely not active part in NAT. It will however be affected by it, much in the same way as any other device on your network.
UPnP is a bunch of related protocols meant to allow network devices to discover, use and potentially manipulate each others services.
The only protocol Pi-hole cares about is DNS (and DHCP if acting as DHCP server).
Pi-hole is not directly concerned with any of UPnP related protocols, some of them leveraging HTTP.
However, Pi-hole will be asked to resolve a domain as contained in a HTTP URL, and it may be configured to block it, which may prevent UPnP requests to succeed.
But there's normally a (good) reason why someone puts a domain on a blockllist.
It's probably safe to enable and(!) confine UPnP within the limits of your home network, but it's really a convenience feature (for manufacturers, mostly) that poses some larger security risks, e.g. it lacks authentication by default, meaning a UPnP device may accept and process commands regardless of origin. For that reason (and others), you're not very likely to find UPnP enabled in a company network.
I'd rather go the hard way and configure forwarding rules myself, much the same way as I wouldn't hand my house keys to my favourite pizza delivery service just to spare me from having to walk to the door.
Granted, they are hard to come by at times, as even official game support seems to lack full and proper information - as said, UPnP is a convenience feature for manufacturers, mostly.