Certainly the Empress Frederick (her aunt) thought she had become far too imperious, Queen Victoria had warned her to not be too proud and had reminded her of the need to gain the love of her subjects.
The Empress Frederick was rather inclined to be critical of her Hesse nieces - they weren't as pretty as they were cracked up to be, etc. etc. Possibly there was a bit of jealousy at their preferential treatment by Queen Victoria. Alexandra and she were never particularly close anyway and she may have interpreted a certain reserve as pride. Queen Victoria's comment about not being too proud was made in the context of the magnificent jewellery Alexandra had received as betrothal and wedding presents as temptations potentially leading to a swelled head, not a personal admonishment.
Within the British royal family, Alix's exact contemporary was Marie Louise of Schleswig Holstein, who always was a great friend and remained so. She never said anything about Alexandra being proud or puffed up; her main concern was that Alexandra was prone to introspection and melancholy, and she said to her face that if she was like that when she had nothing to worry her, what would she do when she had real troubles? Queen Mary and the Wales princesses were in a slightly older age bracket so weren't particular friends - as was natural in those days, when there were sharp divisions between 'girls' and 'young ladies', so you tended to pair up with your agemates - but I've never read of their being particular enemies either. I would have felt that there would be a certain fellow feeling between Queen Mary and Alexandra as the victims of overbearing and possesive sister mothers-in-law!
While I think 'The lost prince' was mildly interesting, I don't think Poliakoff was particularly insightful about individuals and there was too much 'broad brush' stuff - such as the 'no sympathies' comment - which is not really borne out by contemporary evidence.