Well, as I said before, I think it's important to be upfront with my bias towards Wildshape and granting other classes iconic class abilities via archetype. That said, I do think it is an interesting challenge to design a mechanically balanced Monk archetype around this feature, so I'll give it a shot. For comparison sake, I am going to only try and use the PHB as my reference for balance. Looking at the initial entry into a monk archetype, they primarily grant new ways to use Ki points. Open Hand monks get new options that go along with Flurry, Shadow monks get a cantrip and a number of specific, thematic 2nd level spells that cost Ki points, and Elemental Monks get to choose 2 spell-like abilities that cost Ki points. [EDIT: After the first level of the archetype, I stopped comparing to the core PHB monks.] I think it is also important that we try not to step on the toes of the druid too much. So I feel like wildshape-like abilities will be the entirety of the 3rd level archetype ability, will use Ki points, and also will be more time limited. Additionally, there is the problem that if you base the wild shape ability off of Ki points, a monk will be able to use wildshape more often than a druid. This can lead to a problem with the monk having a ridiculous pool of hit points, which is not truly a druid problem until they hit their level 20 capstone ability. So these things are worth considering. Anyways, here's a really... really rough attempt for your monk subclass.
Way of Tooth and Claw
Wild Shape
Starting at 2nd level, you can use your bonus action to magically assume the shape of a beast that you have seen before. When you choose to use this ability, you can spend one ki point to change into a small beast of CR ¼ or lower. If you spend 2 ki points, you can select a small or medium beast up to CR ½. The form that you select cannot have a swim or fly speed.
You can stay in a beast shape for one minute. You then revert to your normal form unless you expend more ki points for use of this feature. You can revert to your normal form earlier by using a bonus action on your turn. You automatically revert if you fall unconscious, drop to 0 hit points, or die.
While you are transformed, the following rules apply:
• Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours. If the creature has any legendary or lair actions, you can't use them.
• When you transform, you assume the beast’s hit points and Hit Dice. When you revert to your normal form, you return to the number of hit points you had before you transformed. However, if you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form. For example, if you take 10 damage in animal form and have only 1 hit point left, you revert and take 9 damage. As long as the excess damage doesn’t reduce your normal form to 0 hit points, you aren’t knocked unconscious.
• You can’t cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form. Transforming doesn’t break your concentration on a spell you’ve already cast, however, or prevent you from taking actions that are part of a spell, such as call lightning, that you’ve already cast.
• You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can’t use any of your special senses, such as darkvision, unless your new form also has that sense.
• You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. Worn equipment functions as normal, but the DM decides whether it is practical for the new form to wear a piece of equipment, based on the creature’s shape and size. Your equipment doesn’t change size or shape to match the new form, and any equipment that the new form can’t wear must either fall to the ground or merge with it. Equipment that merges with the form has no effect until you leave the form.
Improved Wild Shape
Beginning at level 6, you can now choose a beast form that has a swim speed.
In addition, you can use your ki to duplicate the effects of the alter self spell. As an action, you can spend 2 ki points to alter self without providing material components.
Elemental Wild Shape
Beginning at 11th level, you can now choose a beast form that has a fly speed. Adopting a beast form with a fly speed increases the normal ki point cost by 1.
In addition, you have learned to charge your unarmed and natural attacks with elemental power. As a bonus action that costs 2 ki points, you can choose acid, cold, fire, lightning, or poison. For the next minute, all your attacks deal an additional 1d4 damage of the energy type chosen.
Reactive Wild Shape
Beginning at 17th level, you can now choose to spend 4 ki points to wild shape into a large beast up to CR 1.
If you are in your normal form and you would take damage from an attack you can see, you can use your reaction to use your Wild Shape. You still must pay the normal Ki point cost for the beast form you assume.