As RAW, no, it does not
Being considered a size higher for carrying capacity and push, drag, or lift force is not the same as being a size higher for all purposes, including grappling.
You count as one size larger when determining your carrying capacity
and the weight you can push, drag, or lift.
As RAI, maybe
The loop hole is that the PHB has no reference on why and how the grapple-er can move without speed penalty if it is two size higher, and the DMG does not touch the subject. There are two potential scenarios for why a grapple-er one size or lower has it speed reduced, though; the grappled creature have enough leverage to make the movement difficult or that the grapple-er cannot carry comfortable enough the grappled creature as in "difficult terrain", and that the weight of the creature is the deterrent.
In the first case, since Powerful Build does not increase per-se the size, it is safe to assume that it does not help to increase the comfort or reduce the leverage and, such, the speed is halved.
In the second case, where is the weight that maters, things change. Powerful Build improves the carrying, dragging, and lift capacity, therefore the grapple-er creature does not have its speed halved.
Which one it is
The only evidence I found is in favor of is that weight is the reason for the reduction in speed. The first part is in the text of moving a grappled creature (emphasis mine).
When you move, you can drag or carry the grappled creature with you,
but your speed is halved, unless the creature is two or more sizes
smaller than you.
As far as I can tell, there are no rules on how you drag or, particularly, carry a grappled creature. You can carry over you head a grapple creature as if it were a sack of potatoes. In PHB 176 there is a segment that shows how can you drag, carry and lift something in particular and, as shown, it is a STR and weight contest in which Powerful Build should work as intended.
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up
to twice your carrying capacity (or 30 times your Strength score).
Bottom line
At the end is the DM that has the last word. Talk to your DM and present the evidence. If you are the DM decide what it is best for your campaign. If a particular decision is more powerful that you expected it is in your right to take back the decision.
This is one of those edge cases where a DM would need to make a ruling.
Given that the Tarrasque is exceptionally large, it would be reasonable for the line
This reduction decreases its size by one category
to take precedence over the line
The target's size is halved in all dimensions
In other word, the Tarrasque is reduced by more than half such that it can be considered a Huge creature, rather than still a Gargantuan creature. Given that magic has been used to reduce the Tarrasque (meaning that presumably a resource has been expended and the Tarrasque has failed a save, etc), it seems fair for the Tarrasque's opponents (presumably the party) that this should be the case.
Of course, it may also be reasonable for a DM to say that the Tarrasque is so large that reducing doesn't reduce its category, so the lines above take precedence the other way around, leaving the reduced Tarrasque as still being a Gargantuan creature.
Ultimately, without a Word of God answer, this one is up to the DM.
Best Answer
Rules-as-written
You have referenced the rules correctly.
From the spell (emphasis added):
From the DMG:
There is nothing in the rules that says this won't work.
If you are a player
You can anticipate gleefully running through dungeons, banks, palaces, reducing doors as you go. Don't stop there. You can enlarge them, too, and see what happens. Reduce the stones in the walls of the castle and watch it fall, the wheels on the wagon and watch it crash, any number of other things. Reduce the throne under the queen, the tree the bird is perched in, the witch's caldron, the possibilities are endless.
Only you might want to work it out with your DM ahead of time. You can decide something will work a certain way by reading the rules, but you only really know how it will work when you try it in the game and the DM narrates the results of your actions.
If you are a DM
If you're happy with this use of enlarge/reduce, go for it.
If not, don't. Allowing enlarge/reduce to open any door may be contrary to what you want in the game. If it's an issue, feel free to customize it.
Customizing the rules for your game is not straying from the rules. Those are the rules. The DM is the final arbiter of how things work in the game.
I have not dealt with this particular rules challenge as a DM, although I've dealt with similar ones. A reasonable and simple modification is to require a skill check whenever an object is being enlarged or reduced. Most objects, set the DC so low as you not even need to check it. For objects that have some resistance, set a higher DC.
You could also simply rule that reduce/enlarge does not work on attached objects.
Depending on your specific game and players, you might find you need to fine tune your house rule to make sure it works for your table.
Some players may want to argue with you about it. I refer you to "Rules Discussions" in the DMG:
Usually it's less fun to have lengthy rules discussions in session. I find it's better to make a ruling and move on.