You can get trekking shoes (good quality) on rent from hippie in hills in Manali (which is generally the starting point for Hamta Pass trek). Hippie in hills - 8978614753 (if the number does not work, you can try checking with them on their Instagram handle - they'll reply)
They are available for 300-400 per day, however if it does not fall in you budget, you can get good quality gumboots (waterproof) as well from small shops on the way to Solang / Gulaba - you'll get those for INR 100-150 per day. Ive used both - even the gum boots are fine for snow / rain trekking