This is a pic of the dispenser drawer. Not sure what the opening on the top right is.

Every washing machine I've had with a bleach opening very clearly says in the manual to not put anything in the bleach opening except bleach. I don't know why or if it would cause a problem so I always follow the manual.

Bluing works on the principle that the color blue appears brighter...bluing will not whiten. It can also make things duller. Mrs Stewerts seems to be all I see in the stores now days...I read that it's used as ink...regardless this stuff will cause permanent staining and must be well diluted and added to the water in the washer not dumped onto clothing. I can't see how it would be safe to use it in a front loader...there are better options if it's whiter whites your looking for. Pro Supply Outlet sells pure Sodium Percarbonate (oxygen bleach) it's completely scent free and will work like magic to whiten your whites. Depending on how dingy your whites are it may take a couple washes...There's no bleach activator in it so you need to use hot..the hottest water your machine uses.

Back to bluing...it was commonly used in professional laundries as a finishing treatment but only on already white whites. Our grandmother's also often used bluing periodically but only on non dingy whitees. Your not alone there's lots of folks that think bluing will whiten dingy whites...it wont. Lol.

Bluing literally gives the fabric a **very faint** blue tinge which causes dingy whites to appear brighter white by counteracting the slight yellowing and greyness.

Laundry Bluing

Oxi White Revive powder is excellent.

https://acaai.org/allergies/types/allergy-myths/chlorine-allergy

...You can’t be allergic to chlorine, but you can be sensitive or have a reaction. If you are sensitive, should you stop swimming? No, instead you should find and treat the cause of your sensitivity and find relief.

What you think is a chlorine allergy might actually be underlying asthma, exercise-induced bronchoconstriction (EIB) or bronchospasm. Your runny nose might be due to other underlying allergy problems...

I am one of those people who developed an issue after long term exposure. I was a vet tech in my youth, during the time that parvo appeared. I spent years in kennels having to douse everything in bleach. Then I would come home and have to spray my shoes and wash everything in bleach in order not to spread it to my personal dogs. Eventually it got to the point that the smell of bleach would cause my eyes to start to water and swell, nose itch, sneezing and finally wheezing. So my allergist told me to avoid exposure for a while, I ended up having to leave the vet care industry for 10+ years. I eventually opened my own kennel and was thankful that in the interim bleach free products for disinfecting against parvo, distemper and lepto became available. But I still avoid bleach. Oddly enough I just had to have a full allergy skin panel done for foods. I discussed the bleach thing with the dr and she said that my reaction had minimized but that I shouldn’t assume it wouldn’t be exacerbated if I just started using bleach with abandon.

There are also products that contain "optical whiteners"...essentially white dye. The product is deactivated by chlorine bleach. Dacron fiber is naturally a dingy color....so they are treated with white dye. That's why many wash and wear things say not to use bleach.
Not sure what products currently contain those dyes....but Wisk used to.....and I am allergic to Wisk....but not to chlorine