I would recommend you to convert your global portfolio returns into US-$
Kenneth French provides several global factor-returns for the entire global stock market of developed countries. The description states, that
all returns are in U.S. dollars, include dividends and capital gains, and are not continuously compounded.
[...] The global portfolios use global size breaks, but we use the B/M breakpoints for the four regions to allocate the stocks of these regions to the global portfolios. [...]
So in fact, before calculating size- and book-to-market breakpoints, all these measures are converted into US-\$ instead of using local currencies. Therefore, i recommend you to convert your daily portfolio return into US-\$ (simple currency conversion) and apply your regression approach with US-$ portfolio returns. This is a common approach in international empirical research, as your results are based from the view of a US-investor and therefore your results are comparable to US-studies.
Furthermore, if you just convert the factor-returns from US-\$ into Euro, you would omit the fact, that the associated breakpoints and therefore the portfolio sort is based on variables measured in US-\$. This would bias your factor-exposure, so results are quite useless and not easily to interpret.
For local studies, it is often seen in some papers, that you use portfolio returns in local currency and replicate the Fama-French methodology for the specific country, i.e. you calculate your breakpoints in local currency and derive local factor-returns. However, for a global stock portfolio, this approach is quite inconvenient.
You may also take a look at this related question.