This paper focuses on the full-duplex Gaussian two-way diamond relay channel (TWDRC), where two users wish to exchange their messages with the help of two relays, when one relay performs a lattice-based Compress-and-Forward (CF) scheme. The other relay performs either Decode-and-Forward (DF), Amplify-and-Forward (AF) or a lattice-based CF scheme. We start by proposing a novel lattice-based CF/CF scheme, where both relays perform CF. As opposed to the single relay case, two main challenges arise: the quantization rate region is constrained by two Multiple Access Channel (MAC) rate regions, and each user has to combine two noisy observations to decode the sent message. These challenges are tackled by exploiting MAC techniques and Maximum Ratio Combining methods, which are used to combine the two noisy observations. We then characterize a general achievable rate region when one relay sends a version of both users' messages (eventually noisy), which encompasses both AF and DF, and the other relay performs CF and derive the corresponding achievable rate regions. Finally, the three proposed relaying schemes are compared via numerical evaluations. Clearly, the position of both relays has an impact on the best relaying strategy.