I have some experience I can share with you, but I wouldn't limit your options to only those two countries.
In Tajikistan you have the Pamir ibex and the Bukharan ibex depending on location and what record-keeping organization you may belong to. You may be driving just a few hours from Dushanbe, or you may transfer to different parts of the country, some needing 18-hour transfers, like most of the sheep areas. Horses are used in Tajikistan, but not as extensively and more hiking is generally required. Quality can be good, and you can hunt the peak rut in December when your chances are higher at quality animals. On average, trophy quality is "good" not as good as what you will find in Kyrgyzstan or Kazakhstan. Although some ibex in the high 40's (or possibly larger) are taken, most will be in the low 40's. Altitudes range from about 12k up to 14k depending on area, even higher in the Pamirs and sheep areas. Pricing can be among the lowest of any Asian country for ibex starting around $6k and going up from there, but you cannot take trophies home as checked luggage, and some outfitters are more inclusive with their pricing than others. When you add $2,500 or so for trophy shipping (including USA broker fees), I think you will get a better value in Kyrgyzstan, and likely a bigger ibex. Tajikistan does offer some world-class wild boar hunting. that can be added before or after an ibex hunt.
Mongolia has Altai/Siberian ibex in the far west, Gobi ibex in the south, plus ibex found in the Hangai mountain. The Hangai ibex are sold by some companies are "big Gobi," while others sell them as a third species, the Hangai. Altai ibex average in the low to mid 40's, but there are special areas for bigger trophies if you are willing to pay a higher price. True Gobi ibex average about 35" and combine great with White-tailed gazelle, Hangai average about 40". Mongolia is a bit more expensive but pay close attention to what is included and not included with each outfitter. You will likely have you pay for additional expenses with most outfitters including domestic transfers by air or car to each hunting region, an extra cost for your rifle permit, and extra cost for the cites and export certificates, plus trophy shipping. Mongolia is a great destination, from a hunting perspective as well as a touring/cultural perspective. The altitude is lower (usually 8-10k in the Altai and even lower in the Gobi) and hunting is by 4wd jeep and horses primarily (we also used camels to pack out animals on my hunt).
Keep in mind you still have Kazakhstan to consider, which is a good option for quality ibex and maral trophies, but at a premium price point and a pay-by-size trophy fee scale. When you add up the trophy fees, trophy shipping, rifle permit, cites permit, and ground transfer fees, Kazakhstan will be quite a bit more in price, comparable to Mongolia. Generally speaking, the best ibex areas do not have good maral, and vice versa. The best ibex areas have a mid to high 40" average with trophies over 50" taken annually, but remember, trophy fees are priced by size here. Kazakhstan does offer Siberian roe deer, which can be hunted closer to Almaty if not available in the ibex area.
In my opinion having hunted most of these countries myself, I think Kyrgyzstan if a very good choice for a first Asian ibex hunt. You can take your trophy home as checked luggage and the hunt prices in Kyrgyzstan are largely all-inclusive. Quality with the better outfitters who have exclusive control over their concessions will average 45-46" with realistic chance at 50"+. Altitudes average in the 12K range, some hitting 13 feet, but you have extensive horseback use in this country. You can reach most areas in 10 hours or less by car from Bishkek.
Pakistan is another great destination for either the Himalayan or the Sindh ibex, not only for the hunt, but also for culture, touring, and country itself. This is an adventure destination, and although safe, I am not sure I would recommend Pakistan for your first Asian hunt as parts of the experience can be a bit overwhelming in certain areas. The altitudes and terrain can be extreme here and there are no horses. Instead, yaks are offered as assistance. From my experience, the yaks are easy to ride but slow. The porters will carry all of your gear on these hunts, you only need to carry a walking stick. The mountains here are the most spectacular I have ever seen.
Turkey is a great option, but increasingly expensive, and fewer permits create additional competition between outfitters at the license auctions each year. Pricing can also be very expensive as the tags for the best trophy-producing areas command big dollars. All that said, you can still find hunts priced in the teens that will give you a great experience, mature trophy 9+ years of age and a great overall hunt. Beautiful ibex, and some of the best guides I have hunted with. Turkey also has some of the best wild boar hunting and other species, but most of the other species will have steep price tags (Anatolian red deer, Anatolian chamois, urfa gazelle, etc).
Russia has Siberian ibex at attractive pricing, but I personally would avoid anything to do with this country at the time being. Because it is listed as a category 4 "do not travel" advisory from our Dept of State, you don't have the option of buying travel insurance, which is a significant risk if things change for the worse. And with financial sanctions, you won't be able to send funds to a Russian outfitter unless that outfitter has a foreign account. Furthermore, I hear trophies from the past few seasons are not leaving the country, but I haven't heard exactly why (could be due to economic sanctions?). Most importantly, I do think you would have a higher risk of an incident when in Moscow as the wrong customs officer could potentially look for any reason to detain an American. Besides, my own values won't let me patronize Putin's war effort, even on a very micro scale, but that is just me. Too bad, as majority of the Russian people and outfitters probably want nothing to do with Putin's war or the kleptocracy in power.
Azerbaijan does not have ibex, but they have Dagestan tur at pricing similar to what you are shopping for. Sadly, the country closed over covid and never reopened. This is a great hunt, and Baku is a neat city right on the Caspian Sea. If it does reopen, I suspect there will be a price spike. Only time will tell. I took my tur home as checked luggage, but this was no longer an option at the time the country closed.
Have you considered Spain? Spain is more of a gentleman's hunt, but you still earn your trophy. Fantastic people, fantastic food and touring, and you can find hunts to fit almost any budget, form $6k "any size" ibex hunts on private leases to very expensive Gredos hunts inside National Park areas for super-gold medal class trophies.
Another option I recommend to anyone who enjoys mountain hunts are European chamois. They are underappreciated and often very affordable and they make great combination hunts with other species. You can hunt them from the smallest Cantabrian subspecies in Spain to the largest, the Carpathian in Romania and often for much less than any other mountain game animal.
Feel free to send me a PM if you have specific questions.