This paper conducts compressive strength tests on foam concrete prepared under four factors and three levels through the design of orthogonal experiments. It delves into the phase change rules of the load–displacement curves obtained under various mix proportions. Furthermore, based on the 1-day and 3-day compressive strength values, the study explores different mix proportion results using range analysis and variance analysis methods, thereby determining the optimal mix proportion that can satisfy the maximum 1-day and 3-day compressive strength values. The results indicate that the compression process of rapid-hardening foam concrete includes four stages: initial compaction stage, elastic stage, yielding stage, and plateau stage, with each stage having different causes. Additionally, the sensitivity sequence of factors affecting the 1-day and 3-day compressive strength of rapid-hardening foam concrete is respectively rapid sulfoaluminate cement (α) > water-reducing agent content (δ) > foam content (β) > water-cement ratio (γ) and rapid sulfoaluminate cement (α) > water-cement ratio (γ) > foam content (β) > water-reducing agent content (δ). With 100% sulfoaluminate cement content, the 1-day and 3-day compressive strength values can reach 1.7054 and 2.5471 MPa, respectively, which are 13 times and 7 times the minimum values of 1-day and 3-day compressive strength under other admixtures. The analysis shows that the content of rapid sulfoaluminate cement has the most significant effect on the 1-day and 3-day compressive strength of rapid-hardening foam concrete, with foam content having the least impact on 1-day compressive strength and water-reducing agent content having the least impact on 3-day compressive strength. By integrating range analysis and variance analysis, the optimal mix proportion that simultaneously satisfies the maximum 1-day and 3-day compressive strength is determined to be 100% content of rapid-hardening sulfoaluminate cement, 4% foam content, 0.55% cement ratio, and 0.12% admixture content. Overall, this study provides theoretical support for the research and development of new rapid-hardening foam concrete materials and has significant practical implications for the emergency repair and construction of infrastructure projects.