I don't understand this warning. I fitted the exact same battery, the Exide ytz14, to my T1050, and had no problem. The dealer had filled it himself and sealed it, and he knew it was going to go on its side, but still gave me a 2-year guarantee.
As you say, it's an AGM battery, so you should have no problem. But if you're concerned, best UK price I've found on Yuasa batteries is from MDS batteries - they supplied one for my TS, and fast too. For the Tiger 1050 it is: https://www.mdsbattery.co.uk/yuasa-ytz14s-motorbike-battery.html
NB. They do say that: Please be aware that if this battery is mounted on its side or at more than 30 degrees from vertical in your bike then you must leave it to stand in an upright position after you have filled it for at least 24 hours before you install it.
Definitely worth upgrading your battery to the best you can get. Letting the starter grind too slowly over time risks destroying your starter mechanism (a very, very expensive item to replace - friend on this forum had to do it).