You're going about this wrong
You don't want to take undersized mount. You don't want to have to boost your T-Rex's strength score. You want to ride a T-Rex. Fortunately, society rules let you do that:
How many animals and companion creatures can I have at any given time?
At the beginning of each adventure, you may select one companion creature to be your active pet. This creature may be a class-granted creature, such as an animal companion, familiar, eidolon, or phantom, or it may be a creature that you purchase, such as a combat-trained bison. This creature may participate fully in combat, skill checks, and other challenges in the scenario. In addition to the active pet, you may bring up to two additional creatures. The first is a mount or beast of burden, such as a horse or mule, which may only take move actions. The second is a familiar or mundane pet that does not participate in combat or other challenges. Such familiars grant their basic special ability, such as a bat's +3 bonus on Fly checks, the Alertness feat, and access a witch's spells. They do not grant other bonuses, such as a wysp's resonance ability (Pathfinder Roleplaying Game: Bestiary 5 282).
The way to do this is to release your animal companion to take another on, brought along as your 'beast of burden'. Animal companion T-Rex's have the animal companion stats, while not-animal-companion T-Rex's use the not-animal-companion stats. You can release your animal companion in PFS at any time, and as a hunter replacements are free at the end of a scenario. This means you can't teach your T-Rex extra tricks beyond those it starts with, but that should be okay until you're a high enough level that your animal companion T-Rex stats aren't garbage.
At the beginning of each scenarion you release your companion from companion status and rely instead on Handle Animal to keep the T-Rex on your side. It's a Gargantuan creature with 32 strength, so it can carry you (and your party) without effort as it possesses a light carrying capacity of 5,536 pounds. At higher levels, the companion stats are better, but it will take a good long while to get there. After releasing the T-Rex, you take the other animal you brought (probably a cat, cause they're really cheap and animal companion-able), companionize it, feed it to your T-Rex or otherwise kill it off, and proceed with your evolution points and other features applied to yourself.
As a result, you don't need undersized mount. You need methods of dealing with your incredibly dangerous dinosaur ally. In this regard you have several resources:
1) Tricks. Your T-Rex enters play with 1 trick at level 1, and gets a second trick at level 3. You can also add one trick by making a successful training check once per scenario, which (since you have to acquire a new, different T-Rex after each session) means one more trick, potentially, at each level. 'Down' is definitely the most useful trick here, followed by 'Come' or 'Heel'. Additional tricks can be spent on whatever best serves your needs for the specific scenario at hand (since you'll have a mechanically new T-Rex next time).
2) Handle Animal checks. With a DC 25 check, you can "push" the T-Rex to keep it out of trouble. To achieve such a check at level one with any sort of reliability, you're probably gonna need to spend resources on increasing your Handle Animal skill.
3) Wild Empathy. As a last-ditch measure, you can try to use Wild Empathy to save you from your T-Rex. A successful check can improve the animal's attitude, preventing it from deciding to eat you, but the DC is 20 to 25 and your bonus is just level+Charisma.