Hi and welcome! You bag is gorgeous and looks to be in very good condition.
I'm not familiar with the conditioner you are using. Many of us here use lotion like conditioner first and apply several layers, and then 'finish' with a thicker, waxy conditioner in a layer or two to seal in moisture. Forum favorites are Leather CPR for the lotion style, and then Blackrocks or Renapur for the waxy. Some conditioners are known for darkening leather; we try to stay away from those in general. Is your conditioner lotion like or more solid/waxy?
Dry bags do often require several applications. Dunking can also help very dry leather as it soaks in the water (and then you begin applying conditioner before the leather completely dries, locking in moisture). And, some areas often look darker when conditioned. This *usually* fades as the bag dries and can take several days. Straps are also tricky because they tend to be the dirtiest and oiliest part of the bag due to hand oils. I find straps tend to be a shade or two darker on many bags because of this and don't consider it an issue unless it's noticeable to non-bag people

People with more experience than I will chime in soon. You may want to take pictures of the strap for us to look at.
I would ask myself if the strap is getting tacky/sticky, or if each conditioner application is being fully absorbed. If you are using a lotion style conditioner and it's being fully absorbed and seems to be improving the leather, I would apply another coat or two. If it keeps darkening the strap I would stop and work on the bag and see if it is also thirsty and darkening with each application. In the end you want your bag and strap to mostly match and be equally hydrated, so work with an eye toward making that happen. Really gorgeous backpack!