I've stayed at most of the MAN hotels as I am completely neurotic about missing a flight and insist on staying on site the night before for anything departing before 11 am. (I also hate early starts with a passion)

The Hilton DT, Clayton and CP are all 10 minutes or so walk to T3, or have shuttles you can pay to book. The new HI is very nice, and handy for T2. If you don't want to risk getting wet, I think the Radisson is the only option, though it's very overpriced. That said, all MAN hotel prices have been silly since the pandemic, though if you can use an Amex cash back offer and need nights to trigger status/benefits they can be a more reasonable option. DT is a bit faded but you can use digital key to avoid the queues and pop back when it's quieter for cookies and extra water!