Children from bilingual backgrounds benefit from interactions with parents and teachers to re-acquire their heritage language. In this paper, we investigate how this insight from behavioral study can be incorporated into the learning of small-scale language models. We introduce BAMBINO-LM, a continual pre-training strategy for BabyLM that uses a novel combination of alternation and PPO-based perplexity reward induced from a parent Italian model. Upon evaluation on zero-shot classification tasks for English and Italian, BAMBINO-LM improves the Italian language capability of a BabyLM baseline. Our ablation analysis demonstrates that employing both the alternation strategy and PPO-based modeling is key to this effectiveness gain. We also show that, as a side effect, the proposed method leads to a similar degradation in L1 effectiveness as human children would have had in an equivalent learning scenario. Through its modeling and findings, BAMBINO-LM makes a focused contribution to the pre-training of small-scale language models by first developing a human-inspired strategy for pre-training and then showing that it results in behaviours similar to that of humans.