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If you go down that route I'd wash the lid separately. The seal on my keepcup lids tend to last a couple of months before developing leaks. Caused by me using it for filter coffee (so a temperature around 90 degrees or so as opposed to milk drinks which would be around 50-60). They recommended letting the coffee cool a bit before putting the lid on. I imagine a dishwasher would probably damage it in the same way.
How about a KeepCup? The larger, insulated, dishwasher safe version
https://uk.keepcup.com/shop/double-wall-cups/longplay-pistachio