This paper considers the coded downlink massive random access problem in which a base-station (BS) aims to communicate descriptions of the sources (X1, ·s, Xk) to a randomly activated subset of k users, among a large pool of n potential users, via a common message in the downlink. Assuming that the downlink channel is noiseless, this paper investigates the lossy source coding setting where upon receiving the common message from the BS, each active user aims to recover a reconstruction Xi of their intended source Xi, such that the expected distortion between (X1, ·s, Xk) and (X1, ·s, Xk) is less than D. In this paper, we show that a previously proposed lossless coding strategy and its corresponding codebook construction for exchangeable sources, the urn codebook, can be extended to the lossy source coding setting using the Poisson functional representation. With this coding strategy, we show that for exchangeable sources (X1, ·s, Xk), a common message length of R(D) bits plus an overhead of O(k) bits, independent of n, is achievable, where R(D) is the rate-distortion function for compressing (X1, ·s, Xk). If the sources are i.i.d., this overhead can be reduced to O(log (k)) bits.