Hi
The monsoon apart for a moment, the most popular short trek is a Poon Hill variant, (3 - 6 days, say) and the best short trek, in my view, is Jomsom - Kagbeni - Muktinath because of the scenery and interesting villages (K and Jharkot).
PH is on the southern slopes of the Annapurnas and as such gets a lot of rain (Pokhara is one of the wettest areas of Nepal) so is not likely to be the best option in July. J - K - M is in the rain shadow and gets much less rain so is a great option - if you can get there and back which, as said, has risks. KTM - Pokhara flights can be cancelled in the monsoon and the Pokhara - Jomsom flights can be delayed/cancelled at any time of the year because of the strong wind in the Kali Gandaki valley (flights are scheduled in the morning to avoid the worst of the wind) and the cloud cover adds a further source of celays.
Whether you should consider depends on your attitude to risk and the prevailing weather conditions - you could be lucky with a drier than average monsoon.
If you want to play it safe, there are trek options nearer to to KTM and on the valley rim:
The Helambu Circuit is a possibility, most standard itineraries seem to be 7 days KTM to KTM but it caan be done in 6 (note there is a lot of up and down and it's not as easy as ome seem to suggest). The start is at Sundarijal which is only a 45 ride from central KTM/Thamel and 9typically) finishes at Melamchi Pul. MP is a 4-5 hour bus ride back to KTM but this is subject to monsoon dleays too. It took me 8/9 hours one August. Helping to dig out the road and roll the rocks into the river while my p/g looked for a homestay just in case it could not be cleared was fun and part of the experience. This wouldn't be the case if a flight was at risk. There are shorter versions...
Probably the most popular short trek close to KTM is Sundarijal - Chisapani - Nagarkot - Dhulikhel (popular because N and D are mountain viewpoints but you would need to be lucky in July). It's shorter than you are interested in but can be lengthened; start from Budhanilkantha to Chisapani, N, D, on to Panauti, climb Pulchowki and finish at Godavari. No experience of this extension (looking into it as part of the next trip). The advantage of this is you can drop down into the valley and get back to KTM at any point.
Good luck.
scoodly