Could diatom blooms affect albedo of an exoplanet? And perhaps display a seasonal variation if the exoplanet has an axial tilt, providing proof of an Earth-like world supporting at least oceans with microbial life?
First off, I am rather certain that by diatom bloom you refer to algal bloom which is a seasonal change of sea color, at least here on Earth, the only system we have observed yet. Your question is indeed rather hypothetical, but from a astronomy/ physics points indeed an interesting one.
Next, let me summarize the assumptions I am reading out of your question:

You assume an exoplanet that is mainly covered by water. Such a waterworld is not only existing in movies, but also a common toy system in climate science (of Earth).

The exoplanet is not only habitable, but also has some kind of algae living in its ocean. A strong assumption indeed, but for our calculation, it gives us numbers for how the color and thus the emissivity of the planet changes.

Indirectly, you also assume our hypothetical waterworld has an atmosphere. Why? An waterworld exoplanet would otherwise have already quickly evaoprated evaporated into space. In terms of greenhouse calculation later on, we will assume that the atmosphere consists only of water vapor. This means that our hypothetical exo-algae on waterworld would not necessary be aerobic. After all, we are after the physics here.


Could algae bloom affect the albedo of an exoplanet? Definitivley yes, but the question is, how much. And if we could observe that from Earth would be a whole different question.
What do the assumptions mean in numbers?
The albedo of open ocean water (on Earth) is $\alpha_O = 0.06$. Concerning the algal albedo $\alpha_A$, I found e.g.

a poster &quot;Algal blooms: how are they harming models used for climate management?&quot; by Ian Jenkinson
an article John Raven: The possible roles of algae in restricting the increase in atmospheric CO2 and global temperature
another article by H Gordon, M Jacobs: Albedo of the ocean-atmosphere system: influence of sea foam.

In short: Algae do not only change the color of the ocean (which color depends on the involved species), but also may produce foam (with $\alpha_A \approx 1$). The latter seems to have the largger effect since $\alpha_A \gg \alpha_O$. One of the above mentioned papers suggests a 10% coverage of the ocean with foam, an assumption which sounds appealing to me.
Some calculations
For a planet with radius $r$ in thermal equilibrium, the incident radiation equals the black body radiation:
$$ \pi r^2 E (1- \alpha) = 4 \pi r^2 \sigma T_{\rm eqi}^4 \Longleftrightarrow 
T_{\rm eqi} =\sqrt[4]{\frac{E (1-\alpha)}{4 \sigma}}$$
In this formula, $E$ is the solar constant ($E_{\rm Earth} = 1368 {\rm W}/{\rm m^2}$), $\alpha$ is the (overall averaged) albedo ($\alpha_{\rm Earth} = 0.3$) and $\sigma = 5.67 \cdot 10^{-8} {\rm W\cdot m^{-2}K^{-4}}$ the Stefan-Boltzmann constant. For Earth this leads to $T_{\rm Earth,eqi} = -18^\circ {\rm C}$ which shows that we have to include the Greenhouse effect, i.e. that the atmosphere absorbs part of the outgoing radiation. We do that by introducing a  factor $\beta$:
$$T =\sqrt[4]{\frac{E (1-\alpha)}{4 \sigma \beta}}$$
On Earth, all Greenhouse gases (${\rm H_2O, CO_2, CH_4}$) contribute to an average $\beta \approx 0.62$ which leads to a more realistic global average temperature of $T = 15^\circ {\rm C}$ for Earth.
For a water world (without algae) we may use $\alpha = 0.1$ which leads to $T_{\rm WW} = 32^\circ {\rm C}$ which compares to a snowball planet with $\alpha = 0.8$ and $T_{\rm SB} = -63^\circ {\rm C}$.