Inclusion body (IB) is a technique used to express proteins that are toxic to the host cells. IBs are aggregate of highly similar or the same kind of proteins in the cytoplasm formed as stress reaction, resulting biologically inactive precipitated proteins. Such stress reactions can be induced by high temperatures, pH-shifts or occur due to high feeding rates (1,2). Active proteins can be acquired later via purification and mild solubilization of inactive IBs (1).
The second method is to use special host strains. In terms of E. coli, some strains are specifically mutated and selected to withstand the expression of toxic proteins (3). e.g. C41(DE3) and C43(DE3) are developed to enhance membrane protein production (4).
Another method is to express proteins in the periplasmic space. In E.coli, you can achieve that by fusing your recombinant protein with a proper signal peptide, which targets the recombinant proteins to the periplasm via post-translational Sec-dependent pathway (3).
In addition, I also found this product list that provides several solutions for toxic protein expression, including expression in yeasts. But in that case it is similar to your situation so I am not quite sure why it doesn't work out for you.
(1) Slouka C, Kopp J, Spadiut O, Herwig C. Perspectives of inclusion bodies for bio-based products: curse or blessing?. Appl Microbiol Biotechnol. 2019;103(3):1143-1153. doi:10.1007/s00253-018-9569-1
(2) Slouka, C., Kopp, J., Hutwimmer, S. et al. Custom made inclusion bodies: impact of classical process parameters and physiological parameters on inclusion body quality attributes. Microb Cell Fact 17, 148 (2018). https://doi.org/10.1186/s12934-018-0997-5
(3) Rosano Germán L., Ceccarelli Eduardo A. Recombinant protein expression in Escherichia coli: advances and challenges. Frontiers in Microbiology 5, 172 (2014). https://www.frontiersin.org/article/10.3389/fmicb.2014.00172
(4) Kwon, SK., Kim, S., Lee, DH. et al. Comparative genomics and experimental evolution of Escherichia coli BL21(DE3) strains reveal the landscape of toxicity escape from membrane protein overproduction. Sci Rep 5, 16076 (2015). https://doi.org/10.1038/srep16076