In the last 20 years, more than a dozen quantum paramagnets have been identified to show properties consistent with the formalism of Bose-Einstein condensation (BEC) when a magnetic field is used to induce XY-type AFM order at cryogenic temperatures. Among them, BaCuSi₂O₆ orders between critical fields H_c1 = 23.5T and H_c2 = 49T, below T_c,max = 4K. However, BaCuSi₂O₆ additionally undergoes a structural phase transition at 90K into a monoclinic lattice at lower temperature causing a spatial modulation of the spin gap along the c-axis. This spin gap modulation is suspected to be the reason for a dimensional crossover in the magnetic excitations spectrum observed in the critical exponent of the phase transition close to H_c1. We discuss experimental results on newly synthesized single crystals BaCuSi₂O₆ doped with 10% Sr, structurally healed, which do not exhibit a phase transition at 90K and allow the investigation of the BEC close to H_c1 in the absence of spin gap modulation. We establish the H-T phase diagram based on magnetization, magnetic torque, specific heat and magnetocaloric effect (MCE) measurements in DC and pulsed magnetic fields that address the dimensionality of excitations in the BEC critical exponents.