If you wind up with a more complex process to give the same geometric results as the standard number system, the obvious question is, what value is your number system. You have a few options. You can give up on slopes and say that your number system will not give value to calculations of slope (one of the more common instances where division by zero occurs). Then the obvious question is, when does your division by zero become useful? When will $\dfrac{a}{0}=a$ have some sort of physical meaning that will be useful to a mathematician, philosopher, or anyone else? If this is nothing more than a theoretical construction, is it worth anyone's time? I do not have the answer to that.

Another possibility is to force the math to work with your system. It will still yield the same result as more traditional mathematics when it comes to slope, but it will still use your division. Define $\mathcal{L}(\mathfrak{N}^2)$ to be the set of all lines in two dimensions. Given any line $L \in \mathcal{L}(\mathfrak{N}^2)$ define slope to be $\displaystyle \lim_{\begin{matrix}(x_1,y_1),(x_2,y_2) \in L \\ d((x_1,y_1),(x_2,y_2)) \to \infty \end{matrix}}\dfrac{y_2-y_1}{x_2-x_1}$. This is probably not the best notation, but it is the best I can come up with this late at night. It basically says that you are defining the slope of the line to be the limit as you take the slope between points that are increasingly far apart on the line. For non-vertical lines, it is easy to show that because slope is constant, the limit will equal the constant slope. It is unchanging no matter how far apart the two points you choose are. For vertical lines, you wind up with $x_1=x_2$ no matter how far apart the two points are, and you wind up with the numerator getting increasingly large in the positive or negative direction. In order for this limit to exist, you need to define a single point at infinity (basically causing your numbers to wrap around). For the purposes of slope, vertical lines would wind up with a slope of infinity.

But, this has its own weirdness. If you allow an infinite slope, and you are writing lines in the form $y=mx+b$ where $b$ is the value of the $y$-intercept, can you write $y=\infty x + b$ for a vertical line? Not really, because there is no $y$-intercept. So how about writing the line as $y-y_1 = m(x-x_1)$. This becomes $y-y_1 = \infty(x-x_1)$, which again is a rather meaningless notation. If you cannot really do algebra with infinity, what purpose does it have? Are you basically saying that your number system does not work with vertical slopes?

Whichever way you go, this is not really different from existing (and simpler) algebras that already exist without dimensional units. So, again, the question becomes, what value does your system provide that these other systems do not?

You bring up infinitesimals. There is already a rigorously defined theory that includes infinitesimals. It is the theory of Hyperreal numbers:

https://en.wikipedia.org/wiki/Hyperreal_number
Before we progress any further, perhaps come up with some concrete examples of how your theory might be useful. Without some solid examples of when it is useful to have dimensional units, I cannot justify to myself any further work on this. If you want to continue thinking about it, that is fine, but without some understanding of how it might further mathematical knowledge, it seems like a waste of time.