I would argue that it is not. The major reason for this is that the sizes of domestic markets in the EU vary so widely that it is possible for a country to have a decent GDP per capita even if it is not particularly competitive internationally.
For example, France's domestic market is much larger than that of the Eastern European countries that have joined the EU. This means that French companies have had an advantage over Eastern European ones as they have been able to grow without having to export. Right now, France has a higher GDP per capita, but the Eastern European countries are often more competitive internationally because of their low wages.
So, I would argue that countries can become rich without being internationally competitive and poor countries can have advantages in international competition. GDP and other indicators, like consumer prices and exchange rates, are all only general indicators that provide only a rough approximation of competitiveness. Tracking indicators like GDP show only "changes in relative competiveness" (Mattine Durand and Claude Giorno). Therefore, GDP per capita is not a major indicator of international competitiveness among EU countries.