Grand Duke Adolph did not have the means or the motive to join the German Empire.
Part One, the means to join the German Empire.
Luxembourg did have a constitution in 1890, and it did not, repeat NOT say that the Grand Duke was an absolute, all powerful ruler.
But that was not the important obstacle to Luxembourg joining the German Empire.
The Grand Duchy of Luxembourg was part of the German Confederation from 1815-1866.
The King of the Netherlands was also the Grand Duke of Luxembourg from 1815-1890.
Luxembourg was defined as an independent state and a neutral one in the Treaty of London in 1839. That legal status would have made an attempt by Luxembourg to join another and not neutral country an excuse for war by powers opposed to that.
The Austro-Prussian War of 1866 resulted in the dissolution of the German Confederation and the creation of a Prussian-led North German Confederation in 1867, which was replaced by the German Empire in 1871.
The Treaty of London (1867) reaffirmed that Luxembourg was neutral, which of course implied that it was illegal for Luxembourg to become part of another country which was not neutral. The neutrality and independence of Luxembourg was guaranteed by the powers which signed the treaty:
The Austrian Empire, represented by the Count Rudolf Apponyi
The Kingdom of Belgium, represented by Sylvain Van de Weyer
The French Empire, represented by the Prince de La Tour d'Auvergne-Lauraguais
The Kingdom of Italy, represented by the Marquis d'Azeglio
The Grand Duchy of Luxembourg, represented by Baron de Tornaco and Emmanuel Servais
The Kingdom of the Netherlands, represented by the Baron Bentinck
The Kingdom of Prussia, represented by the Count Bernstorff-Stintenburg
The Russian Empire, represented by Baron Brunnow
The United Kingdom, represented by Edward Stanley, 15th Earl of Derby
So if the King of Prussia and German Emperor accepted an application by Luxembourg to join the German Empire, he would be violating international treaties.
If someone told the Grand Duke of Luxembourg he would stop him from joining the German Empire, and the Grand Duke said "You and what army?", the answer would be the armies of some of the powers that signed the treaty and were not friendly to Germany, powers like Russia and France, for example. An attempt to annex Luxembourg to Germany could have resulted in an earlier World War One.
Part Two, the motive to join the German Empire.
I may note that Adolf Wilhelm August Karl Friedrich (1817-1905), who became Grand Duke of Luxembourg in 1890, also became Duke of Nassau in 1839. The Duchy of Nassau was an independent country, except for being part of the German Confederation. In the Austro-Prussian war of 1866 Prussia fought Austria and the German Confederation. Nassau fought for the German Confederation against the Prussian rebels and was conquered and annexed by Prussia in 1866, dethroning Duke Adolf.
Duke Adolf became an ex monarch, a ruler without a country, and remained one for 24 years until he inherited Luxembourg, as a result of Prussian treason and aggression.
Assuming that Grand Duke Adolf was patriotic for the German Empire and wanted to join it is like assuming that Jefferson Davis, Geronimo, & Queen Liliuokalani were patriotic for the USA, or that Paul Kreuger was patriotic for the UK, or that Louis Riel was patriotic for Canada, or that Rabih az-Zubayr was patriotic for France.