Miller (1981) specifically addressed your question in Chapter 11 of this book:
- Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 978-0-201-04679-3
He wrote
... problems concerning the reflection of light from the moving mirror,
due to their fundamental relationship to both electromagnetism and
thermodynamics, were the theme of many papers.
He cited the following papers by Hendrik Lorentz, who discussed solutions to first order in v/c:
- De l'influence du mouvement de la terre sur les phénomènes lumineux, Vers. Kon. Akad. Wetensch. Amsterdam, 2, 297 (1886). Reprinted in Collected Papers, 4, pp. 153-214.
- On the Reflection of light by Moving bodies, Vers. Kon. Akad. Wetensch. Amsterdam, 1, 28 (1892). Reprinted in Collected Papers, 4, pp. 215-218.
- Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern (Leiden: Brill, 1895). Reprinted in Collected Papers, 5, pp. 1-137.
However, the most important paper was written by Max Abraham in 1904:
- Zur Theorie der Strahlung und des Strahlungsdruckes, Ann. Phys., 14, 236-287 (1904).
Abraham's results are valid for all orders in v/c. However, he only discussed the problem with respect to a frame is which the reflecting surface was moving, and his method based on classical electromagnetism was extremely complicated.
On the contrary, Einstein in 1905 used the methods of special relativity, which allowed him to achieve all those results by using the rest frame of the reflecting surface, thereby significantly simplifying the computation. As pointed out by Miller:
In the three pages of §8 Einstein deduced the transformation equation
for the energy of a light complex, and then solved exactly the
problems of determining the characteristics of plane waves incident on
a perfectly reflecting surface that is in inertial motion with
velocity v, and of calculating the pressure of light on this surface
[§8, 1-2]. Abraham (1904) spent forty pages to solve these problems.
So Einstein only needed 3 pages to achieve results for which Abraham required 40 pages...
Regarding the experiments to which Einstein alluded to, Miller points to those of Lebedev (1900) as well as Nichols & Hull (1901) on radiation pressure, which were also referenced by Abraham (1904).
