We present a detailed study of the phase diagram of copper-intercalated TiSe2 single crystals, combining local Hall-probe magnetometry, tunnel diode oscillator technique (TDO), and specific-heat and angle-resolved photoemission spectroscopy measurements. A series of the Cu-x TiSe2 samples from three different sources with various copper content x and superconducting critical temperatures T-c have been investigated. We first show that the vortex penetration mechanism is dominated by geometrical barriers enabling a precise determination of the lower critical field, H-c1. We then show that the temperature dependence of the superfluid density deduced from magnetic measurements (both H-c1 and TDO techniques) clearly suggests the existence of a small energy gap in the system, with a coupling strength 2 Delta(s) similar to [2.4-2.8]k(B)T(c), regardless of the copper content, in puzzling contradiction with specific-heat measurements which can be well described by one single large gap 2 Delta(l) similar to [3.7-3.9]k(B)T(c). Finally, our measurements reveal a nontrivial doping dependence of the condensation energy, which remains to be understood.