^ This. Also, unfortunately, I had issues with the basic stick-on grips because the adhesive was not as good as expected in the rain. It started peeling after just one week... very disappointing. I eventually switched to a metal bracket instead.

• SmallRig L-Shape Grip for Sony RX100 III IV V VA VI VII
• UURig R017 Metal Extension Grip for Sony RX100 These add a bit of weight but make the camera way easier to hold securely.

Been thinking about your situation... honestly, I would suggest being really careful with just a grip in the rain. I once had a handle loosen up on me and it was terrifying. Since you're on a budget, maybe try the JJC HG-RX100 Metal Hand Grip but definitely add a Peak Design Cuff Camera Wrist Strap too. It saved my gear once when my hand slipped. Safety first, y'know?

Oh man, I totally feel your pain! The RX100 is such a beast of a camera but holding it is literally like trying to grip a wet bar of soap, especially in the rain. I almost dropped mine in a puddle a few years back and it was heart-stopping! Honestly, you gotta get the Sony AG-R2 Attachment Grip right now. It is basically the gold standard for fixing this exact problem. It is just a simple, high-quality rubberized piece that sticks onto the front with 3M adhesive, but the difference is absolutely massive! It gives your fingers a nice little ridge to hook onto so you can actually shoot one-handed without feeling like you are about to lose a thousand dollars. It is super cheap too, usually way under your budget, so you can grab it on Prime and have it before your trip starts next week. If you want something a bit more robust tho, I also really love the SmallRig LCS2438 L-Bracket for Sony RX100. It adds a tiny bit of weight but the extra grip surface is fantastic for larger hands. It makes the camera feel way more like a professional tool rather than a slippery toy. Plus, it has a built-in cold shoe if you ever want to add a tiny mic or light later on. You really cant go wrong with either of these, they make the ergonomics so much better. Seriously, get one of these and you will be so much more confident walking around London! Let me know if you have any questions about sticking the Sony one on, it is super easy but I can give you some tips if you need em.