Mathematical analysis of Marginal Cost
I think you're getting things backwards. Marginal cost (at a certain quantity $q_0$) really only makes sense as the derivative of the total cost function with respect to quantity at point $q_0$:
$$
MC(q_0)=\frac{dTC(q_0)}{dq}.
$$
The interpretation of marginal cost as "the cost of producing the last unit of output or the cost of producing the next unit of output" is based on the (linear) approximation 
$$
\widehat{MC}^+(q_0)\approx\frac{TC(q_0+1)-TC(q_0)}{(q_0+1)-q_0}\quad\text{or}\quad
\widehat{MC}^-(q_0)\approx\frac{TC(q_0)-TC(q_0-1)}{q_0-(q_0-1)}
$$
This interpretation is usually introduced in undergraduate textbooks because of its intuitive appeal, not for its mathematical rigor. Imagine trying to explain the concept of marginal cost as the cost associated with an infinitesimal amount of change in quantity to a freshman who probably has no idea what "infinitesimal" means, let alone the concept of derivative. 

A first problem related to interpreting marginal cost as the cost associated with the previous or the next unit of output is that producing the two units (the previous and the next) may imply different costs. This is illustrated by the formulas $\widehat{MC}^+$ and $\widehat{MC}^-$. Suppose total cost is quadratic, i.e. $TC(q)=q^2$. Then it can be easily verified that $\widehat{MC}^+(q_0)\ne \widehat{MC}^-(q_0)$, for any admissible $q_0$. This is problematic because we want the marginal cost (at a particular $q_0$) to be unique. 

Second, and this relates to your question at the end, the concept of "unit" is somewhat arbitrary. And this arbitrariness makes measures such as $\widehat{MC}^+$ and $\widehat{MC}^-$ unstable as well. To illustrate, continue with the quadratic total cost example. When the unit of quantity is kg, the "marginal cost" at $1$kg--as you would refer to the cost associated with producing one extra unit--is $$\widehat{MC}^+(1)=2^2-1^2=3.$$
But what if we suddenly want to change the unit of quantity to gram (g)? Then the "marginal cost" at $1$kg, or $1000$g, would change as well:
$$
\widehat{MC}^+(1)=(1.001)^2-1^2=0.002001.
$$
To avoid such instability caused by changes in units of measurement, marginal cost is defined as a derivative, not as a difference. 

If you want to measure the change in total cost due to change in some arbitrary quantity, simply apply the first formula in your question. But keep in mind that this cost is not marginal cost as an economist would understand it.