Isotope harvesting from the beam dump cooling loop of the Facility for Rare Isotope Beams (FRIB) offers usable quantities of many isotopes unavailable anywhere else in the world. Many of these co-produced isotopes are of significant interest for research in biomedicine, energy, environmental studies, and materials science, and yet they are destined for disposal. One option for retrieval of these rare isotopes is harvesting from the spent mixed-bed resin in a batch collection mode. Alternatively, Hollow Fiber Supported Liquid Membrane (HFSLM) extraction poses a method of quickly retrieving these rare isotopes while overcoming the challenges associated with the beam dump environment: ultra-trace concentrations (ppt), high flow rates (4-6 L/s), neutral pH environment (6-7), vast mixture of elements, and a high radiation environment (1.3 kGy/hr). After one to two years of operation, the two 50-gallon resin tanks of Purolite NRW3460 mixed-bed resin will be hydraulically removed for transfer to storage tanks for waste shipment, at which point the isotopes will be accessible for harvesting. The lanthanides contained within this waste are of particular interest because their long half-lives and high energy density make them excellent candidates for use in nuclear batteries. 177Lu was chosen as a radiotracer to determine the feasibility of harvesting long-lived lanthanides from the mixed-bed resin. This work describes a method for high recovery of the lanthanides utilizing strong acid recirculation coupled with removal of the extracted lanthanides with an Eichrom DGA cartridge. Ninety-seven percent of dosed lanthanides were recovered from a surrogate system into only thirty milliliters of dilute acid, while utilizing approximately three column volumes of strong acid for elution. V was chosen as a radiotracer to determine the feasibility of using HFSLM extraction to remove short-lived transition metals from the beam dump loop. Part per trillion levels of 48V were successfully recovered from the aqueous feed solution spiked with chemically similar species at flow rates of 6 gallons/hour (450 mL/min.) utilizing 0.15 M Aliquat 336 in 3% dodecanol/dodecane in the HFSLM pores and 0.5 M ammonia in 0.1 M ammonium nitrate as the stripping solution. The chemical extraction efficiency from this matrix was found to be 71% in 60 minutes. This is the first ever demonstration of part per trillion level recovery with HFSLM extraction. 177Lu was chosen as a radiotracer to determine the feasibility of using HFSLM extraction to remove lanthanides from the beam dump loop. Preliminary benchtop liquid-liquid extraction tests showed successful extraction of Lu from a neutral pH feed solution with 5 mM Octyl(phenyl)-N,N-diisobutyl- carbamoylmethylphosphine oxide (CMPO) in a room temperature ionic liquid, 1-Hexyl-3methylimidazolium bis(trifluoromethylsulfonyl) imide [C6mim][Tf2N]; however, upon testing with a Liqui-Cel 1.7x10 mini module membrane contactor, it was discovered that the RTIL degraded the polycarbonate mini-module causing cracking and leaking, and no lutetium migrated to the strip solution. It is hypothesized that the high viscosity of the RTIL prevents permeation into the HFSLM pores.