I had this with my middle DS and it's so wearing. What did help for him is not responding when he's being rude and/or asking him to repeat the request calmly/ask nicely. Also, trying to pre-empt his hunger/thirst/etc before he got into a more wheedly kind of state. I was planning to do the following but it didn't come to this.
I would talk to her about it as a neutral time. She might find it fun to have a bit of fun with it - you could practice asking nicely, including very, very, OTT nicely, like "Pretty pretty please, darling mummy the bestest mummy ever ever in the whole world?" (this is hilarious to most 4yos) and also "practice" asking rudely and see if you can outdo each other and make each other laugh with your rude-asking examples, you know, just so you both know what NOT to do, and then some neutral ones which are just stating the need/request in a calm voice. If you want, do it alongside some kind of visual marker like a traffic light with red/amber/green, or sad/happy/extra happy face. Then over the next week or so, try and track how she asks for things - you can either do it privately or you can actually have the chart visible if you like and put a little sticker on the happy/sad face etc. When she does ask rudely, you can then remind her of the chart and let her have a chance to ask again. Even if you choose not to have a visual chart, make sure you praise her whenever you hear her asking nicely (or neutrally) for anything, and especially if you notice that she starts off being rude and then corrects herself. If you do choose to do the chart do this too, but also tot up the points each day or at the end of the week and give a small reward (or lots of praise/attention) and then you shouldn't need the chart any more. You can continue it for a second week if you want but if it's not helping after 2 weeks then don't drag it out because the cause is something else.
Also this is a bit uncomfortable, but consider how you/DH make requests of DC, especially when your patience is running low or you think they are being too slow/messing around - I realised one of the things with DS is that he was copying our tone and wording, and while it sounded unremarkable from an adult to a child, it sounded shocking when he said it back to us or to his brother, which made me reconsider how we were talking to him, and I made a big effort to drop any exasperated/annoyed tone and be more neutral/polite/encouraging in general, even when I wanted him to hurry up - and I think this did actually help a lot.
It's good you notice it starts a stress reaction in you - that was something I didn't initially realise and I think I was reacting a bit more harshly because of it. Something which helps with this is taking three deep breaths and also having the pre-prepared response (e.g. to ask them to try again, and knowing what to look for/praise for) - it's really hard to come up with a helpful, positive, constructive response in the moment when you are in that stress response.