It is uncommon and feels unnatural to narrate anything in one tense only.
Instead of believing your friend (what makes him an expert?) you might want to pick some contemporary fiction and confirm for yourself what writers (and editors) do.
Something like this is fairly commonplace:
I barely manage to drag myself from sleep when the alarm goes off. The night has been much too short, and the sun feels blindingly bright.
Does the tense switch feel wrong? No. People think of the past and future all the time, so if you write from the perspective of your protagonist, of course that person will remember things that have led up to the present moment and consider the future consequences. A person who lives completely in the present is either mentally disabled or a buddhist monk.
What might feel wrong and what you need to do well is how you integrate your backstory into the present events. When do you narrate it? Does it break the pace of the main plot? Is the backstory so long that it feels like you have switched to telling another story? Stuff like that. Only include as much of the backstory as you need for your main story. And include it in a way that feels natural to the narrative viewpoint.
A third person narrator can deviate farther and more completely from the main narrative, while a first person narrator will always be where he is and perceive and process his surroundings. He will usually not "drop out" of the here and now and relive the past as if in a trance, instead he will think of bits and pieces of the past while dealing with the present. So a full chapter backstory in a first person present tense narrative might indeed feel out of place.
