joinAcross2[
   list1 : {__?AssociationQ},
   list2 : {__?AssociationQ},
   sameTest_,
   key : (_String | _Key),
   jspec : ("Inner" | "Outer" | "Left" | "Right") : "Inner"
] := Module[{
    vals1 = list1[[All, key]]
   },
   JoinAcross[
    list1,
    MapAt[
     Function[
      element,
      SelectFirst[vals1, sameTest[element, #] &, element]
     ],
     list2,
     {All, key}
    ],
    key,
    jspec
   ]
];

In[208]:= joinAcross2[{<|a -> 1, b -> X|>}, {<|a -> 1., c -> Y|>}, Equal, Key[a]]

Out[208]= {<|a -> 1, b -> X, c -> Y|>}

Module[{dataset = list2},
 dataset[[All, key]] = Map[
   Function[element,
     SelectFirst[vals1, sameTest[element, #] &, element]
   ],
   dataset[[All, key]]
 ];
 dataset
]