You can definitely do everything in that timeframe! If you are travelling solo, I’d 100% recommend booking a tour! It is the safest way to travel and you don’t have to worry about transportation or tickets selling out. This trip for example has everything Inca Trail & Amazon Adventure by Intrepid Travel with 5 Tour Reviews - TourRadar - I’ve been eyeing it for myself. Are you an active traveller? then something like this would be perfect!
I agree you could definitely visit Machu Picchu and either the Amazon or Lake Titicaca in 10-12 days!
I also think a tour would be a great idea, especially as a solo traveller., to make sure the logistics are taken care of and you have the time to take it all in and get all the local insights too!
I personally recommend Lake Titicaca, this was a highlight for me - it’s absolutely stunning there and there is something about the sky over the lake at the high altitude that makes you feel you are somewhere special. And you can visit the floating islands of the Uros people which is very unique. I didn’t go to the Amazon in Peru, but did in Bolivia, and also such a sacred place, I would try and do both if you can!